Projections are grouped by use, and ordered roughly chronologically by type. A “standard” parallel/meridian is undistorted, and defines the scale printed on the map. Projections with a pink background are especially useful or common. Only those projections shown with diagrams are based on simple geometric constructions. I have compiled a short list of sources and links for your health and happiness.

PROJECTION REFERENCE
Bill Rankin, 2006 (Almost) All the projections available in ArcGIS. 
WALL MAPS OF THE WORLD For display and thematic maps. 
Name  Author, Date  Example, with relevant parameters  Also known as / [Equivalent to]  Properties  EqualArea?  Used by / [Applications]  
Equidistant Cylindrical  attributed to Marinus of Tyre, c.100 CE  standard parallels: 60° N/S, 30° N/S, 0°  Equirectangular [Equidistant Conic with standard parallels equidistant from the equator]  All merdians are standard, with same scale as the standard parallels No distortion along standard parallels  Mapquest (standard parallels 37.5° N/S) Yahoo Maps (standard parallels 51° N/S)  
Plate Carrée (literally, “plane square”)  known since antiquity renamed “carte plate carrée” by M. Armard P. d'AvezacMacaya, 1863  Plane Chart [Equidistant Cylindrical with standard parallel at 0°]  All merdians are standard, with same scale as the equator No distortion along equator Lat/Lon lines make squares Boundary is a 2:1 rectangle  National Geographic’s online MapMachine  
Werner (in ArcMap, use Bonne)  based on Ptolemy’s 2nd Projection, c.100 CE final form by Johannes Stabius (c.1500), promoted by Johannes Werner (1514)  Cordiform (“heartshaped”) [Bonne with reference parallel at 90°N]  All parallels are standard, with the same scale as the central meridian Parallels are concentric circles about the North Pole No distortion along the central meridian  EqualArea  ♥  
Bonne  based on Ptolemy’s 2nd Projection, c.100 CE final form by Rigobert Bonne, c.1752  reference parallel: 60°N, 90°N, 30°N, 0°  Cordiform (“heartshaped”)  All parallels are standard, with the same scale as the central meridian; parallels are concentric circles No distortion along the reference parallel or the central meridian Special Cases: Werner results when the reference parallel is at a pole. Sinusoidal results when the reference parallel is at the equator.  EqualArea  ♥  
Sinusoidal  used in various atlases, c.1600 renamed “sinusoidal” by M. Armard P. d'AvezacMacaya, 1863  SansonFlamsteed Mercator EqualArea [Bonne with reference parallel at 0°]  All parallels are standard, with the same scale as the central meridian No distortion along the equator or the central meridian — this is the only equalarea projection with equally spaced horizontal parallels Meridians are halfperiod sinusoids  EqualArea  
Mercator  Gerardus Mercator, 1569  Cylindrical Conformal [Lambert Conformal Conic with standard parallels at or equidistant from the equator]  Conformal: local angles are preserved, and local circles are not deformed — at every point east/west scale is the same as north/south scale Distortion is constant along any parallel, and any parallel can be defined as standard Constant bearings (rhumb lines / loxodromes) are straight lines Map extends infinitely North and South  Google Maps (note the dynamic scale bar) [navigating a constant bearing]  
Miller Cylindrical  Osborn Miller, 1942  [modified Mercator]  Pleasant balance of shape and scale distortion No distortion along the standard parallel at the equator Unlike Mercator, this map is finite (but not conformal)  Various American atlases  
Gall Stereographic  James Gall, 1885  No distortion along the standard parallels at 45° N/S  National Atlas of the USSR (1937) Rand McNally Various British atlases  
Times  John Moir, 1965  [parallels from Gall Stereographic, meridians curved to reduce distortion]  Pleasant balance of shape and scale distortion Standard parallels at 45° N/S; distortionfree only where the standard parallels intersect the central meridian  Bartholomew Ltd., The Times Atlas  
Mollweide  Karl Mollweide, 1805 popularized by Jacques Babinet, 1857  homolographic elliptical Babinet  Standard parallels at 40º 44' N/S; parallels are unequal in spacing and scale; distortionfree only where the standard parallels intersect the central meridian Meridians are halfellispses; boundary is a 2:1 ellipse  EqualArea  
Aitoff  David Aitoff, 1889  [expanded Azimuthal Equidistant with central point at 0°/0°]  Pleasant balance of shape and scale distortion Boundary is a 2:1 ellipse No standard lines; distortionfree only at central point  superceded by HammerAitoff  
HammerAitoff  Ernst Hammer, 1892  Hammer [expanded Lambert Azimuthal EqualArea with central point at 0°/0°]  Pleasant balance of shape and scale distortion Boundary is a 2:1 ellipse No standard lines; distortionfree only at central point  EqualArea  
Goode Homolosine  J. Paul Goode, 1923  interrupted to show: land, ocean  [Sinusoidal between 40° 44' 11.8" N/S, Mollweide at higher latitudes]  Developed as a teaching replacement for the “evil Mercator” projection. No distortion along the equator or the vertical meridians in the middle latitudes  EqualArea  USGS Many school atlases  
Craster Parabolic  John Craster, 1929  Reinhold Putniņš P4 (1934)  Standard parallels at 36º 46' N/S; parallels are unequal in spacing and scale; distortionfree only where the standard parallels intersect the central meridian Central meridian is half the length of the equator Meridians are parabolas  EqualArea  
Quartic Authalic  Karl Siemon, 1937 Oscar Adams, 1944  Standard parallel at 0°; parallels are unequal in spacing and scale; no distortion along the equator Following the equatorial Lambert Azimuthal EqualArea, equator is π/sqrt(2) times the length of the central meridian Meridians are fourthorder curves  EqualArea  
Cylindrical EqualArea  Johann H. Lambert, 1772  standard parallels: 0°, 30° N/S, 45° N/S  Lambert Cylindrical EqualArea [Albers EqualArea Conic with standard parallels equidistant from the equator]  Parallels are exactly π times the length of meridians No distortion along the standard parallels Behrmann, GallPeters, and HoboDyer are derivatives of this projection These are the only rectangular equalarea projections  EqualArea  
Behrmann  Walter Behrmann, 1910  [Cylindrical Equal Area with standard parallels at 30° N/S]  note  EqualArea  
HoboDyer (in ArcMap, use Cylindrical EqualArea)  Mark Dyer, 2002  [Cylindrical Equal Area with standard parallels at 37.5° N/S]  note often plotted with South up “Hobo” comes from the first names of Howard Bronstein and Bob Abramms, founders of ODT, Inc. (Amherst, MA mapsellers).  EqualArea  ODT, Inc. Carter Foundation  
GallPeters (in ArcMap, use Cylindrical EqualArea)  James Gall, 1855 Arno Peters, 1967 (presented 1973)  Gall Orthographic Peters [Cylindrical Equal Area with standard parallels at 45° N/S]  note  EqualArea  UNESCO NATO Vatican World Council of Churches  
Van der Grinten I  Alphons J. van der Grinten, 1898 (published 1904)  Pleasant balance of shape and scale distortion Boundary is a circle; all parallels and meridians are circular arcs (spacing of parallels is arbitrary) No distortion along the standard parallel at the equator Usually clipped near 80° N/S  National Geographic (19221988) US Dept of Agriculture (from 1949)  
Eckert I  Max Eckert, 1906  Standard parallel at 0º; parallels are equally spaced; distortionfree only at central point Polelines and central meridian are half the length of the equator  
Eckert II  Max Eckert, 1906  Standard parallels at 55º 10' N/S; parallels are unequal in spacing and scale; distortionfree only where the standard parallels intersect the central meridian Polelines and central meridian are half the length of the equator  EqualArea  
Eckert III  Max Eckert, 1906  [Winkel II with standard parallel at 0° — mathematical average of Plate Carrée and Mollweide]  Standard parallel at 0º; parallels are equally spaced; distortionfree only at central point Outer meridians are semicircles; other meridians are semiellipses Polelines and central meridian are half the length of the equator  
Eckert IV  Max Eckert, 1906  Standard parallels at 40º 30' N/S; parallels are unequal in spacing and scale; distortionfree only where the standard parallels intersect the central meridian Outer meridians are semicircles; other meridians are semiellipses Polelines and central meridian are half the length of the equator  EqualArea  National Atlases of the USSR (1937) and Japan (1977) National Geographic  
Eckert V  Max Eckert, 1906  [Winkel I with standard parallel at 0° — mathematical average of Plate Carrée and Sinusoidal]  Standard parallel at 0º; parallels are equally spaced; distortionfree only at central point Meridians are halfperiod sinusoids Polelines and central meridian are half the length of the equator  
Eckert VI  Max Eckert, 1906  Standard parallels at 49º 16' N/S; parallels are unequal in spacing and scale; distortionfree only where the standard parallels intersect the central meridian Meridians are halfperiod sinusoids Polelines and central meridian are half the length of the equator  EqualArea  
Winkel I  Oswald Winkel, 1914  standard parallels at ~50° 27' 35" N/S  [mathematical average of Equirectangular and Sinusoidal — the generalization of Eckert V]  Pleasant balance of shape and scale distortion Distortionfree only where the standard parallels intersect the central meridian with Winkel’s preferred standard parallels ±arccos(2/π), total map area is appropriately π × the length of the equator  
Winkel II  Oswald Winkel, 1918  standard parallels at ~50° 27' 35" N/S  [mathematical average of Equirectangular and Mollweide — the generalization of Eckert III]  Pleasant balance of shape and scale distortion Distortionfree only where the standard parallels intersect the central meridian  
Winkel Tripel  Oswald Winkel, 1921  standard parallels of component Equirectangular at ~50° 27' 35" N/S  [mathematical average of Equirectangular and Aitoff]  Pleasant balance of shape and scale distortion Unlike other Winkel projections, there are no standard parallels on the final map No point is distortionfree  The Times Atlas National Geographic (since 1998)  
Flat Polar Quartic  Felix W. McBryde and Paul Thomas, 1949  McBrydeThomas #4  Standard parallels at 33º 45' N/S; parallels are unequal in spacing and scale; distortionfree only where the standard parallels intersect the central meridian Polelines are onethird the length of the equator, equator is π/sqrt(2) times the length of the central meridian Meridians are fourthorder curves  EqualArea  US Coast and Geodetic Survey  
Robinson  Arthur Robinson, 1963 (published 1974)  Lengths of parallels, central meridian, and pole lines are arbitrarily tabulated for best visual appearance Parallels are equally spaced between the standard parallels of 38° N/S; spacing decreases towards the poles; no point is distortionfree  Rand McNally National Geographic (19881998) CIA (central meridian 10°E)  
Cube  [Plate Carrée between 45° N/S, abutted Collignon projections for the poles]  There are certainly more elegant cube projections than the one provided by ArcMap; one common approach is to place the poles at opposite corners and use Gnomonic projections for each face  Homemade holiday decorations  
Fuller  Buckminster Fuller, 1946 (cuboctohedron), 1954 (icosohedron)  Dymaxion Map AirOcean World Map [Unfolded icosahedron, with each face in gnomonic projection]  No land is cut No natural “up” direction The edges of each face are standard lines; no point is distortionfree, but overall distortion is low  Jasper Johns (1967)  
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MAPS OF HEMISPHERES AND CONTINENTS Very flexible projections that can be tweaked to give low distortion. (Shown with world or continental maps.) 
Name  Author, Date  Example, with relevant parameters  Also known as / [Equivalent to]  Properties  EqualArea?  Used by / [Applications]  
Azimuthal Equidistant  said to be used in ancient Egyptian star maps fully developed in 15th and 16th centuries  centered on: north pole, paris, new york  Postel (1581)  Constant radial scale from central point (i.e., all straight lines through the central point are standard lines) Distortionfree only at central point  the Cassinis' floor map at the Paris Observatory (1680s) United Nations Emblem Flat Earth Society [easily computing distances or missile ranges]  
Equidistant Conic  based on Ptolemy’s 1st Projection, c.100 CE final form by Joseph Nicholas De l'Isle, 1745  optimize for: world, north america, africa  Simple Conic  Parallels are equally spaced: all meridians are standard, with the same scale as the standard parallels; no distortion along the standard parallels Special Case: Equidistant Cylindrical results when standard parallels are at or equidistant from the equator  
Orthographic (also listed as “The World From Space” in ArcMap)  used by Hipparchus, 2ndC BCE renamed “orthographique” by François d'Aiguillon, 1613  centered on 100°W, 45°N  Azimuthal Orthographic  Distortionfree only at central point  [pictorial views]  
Vertical Perspective (listed as a “World” projection in ArcMap)  used by Matthias Seutter, 1740 (observer at ~12,750km) various “farside” projection points used in 18th and 19th centuries  centered on 100°W, 45°N height of observer: 35,786km (approximately a geosynchronous orbit)  “NearSide” Vertical Perspective (Stereographic and Gnomonic being “farside” projections)  Distortionfree only at central point  Google Earth US Weather Service (for satellite data) [pictorial views]  
Lambert Conformal Conic  Johann Lambert, 1772  optimize for: world, north america, africa  Conformal: local angles are preserved, and local circles are not deformed — at every point east/west scale is the same as north/south scale Distortion is constant along any parallel Map is infinite in extent Special Cases: Mercator results when standard parallels are at or equidistant from the equator. Polar Stereographic results when both standard parallels are at one pole.  The American Oxford Atlas Rand McNally National atlases of Canada and Japan USGS (since 1957)  
Lambert Azimuthal EqualArea  Johann Lambert, 1772  optimize for: north pole, north america, eastern hemisphere  [polar version: Albers EqualArea Conic with both standard parallels at a pole]  Distortionfree only at central point  EqualArea  nationalatlas.gov (with central point at 45°N, 100°W)  
Lambert EqualArea Conic (in ArcMap, use Albers)  Johann Lambert, 1772  standard parallel at ~48° 35' N (for Lambert’s own cone constant of ⅞) optimize for: world, north america, europe  [Albers EqualArea Conic with one standard parallel at a pole]  No distortion along standard parallel  EqualArea  superceded by Albers  
Albers EqualArea Conic  Heinrich Albers, 1805  optimize for: world, north america, africa  No distortion along standard parallels Special Cases: Lambert EqualArea Conic results when one standard parallel is at a pole. Polar Azimuthal EqualArea results when both standard parallels are at one pole. Cylindrical EqualArea results when both standard parallels are at or equidistant from the equator.  EqualArea  USGS (standard parallels at 45.5°N, 29.5°N; central meridian at 96°W) US Census FactFinder  
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NAVIGATING AND MEASURING LARGE AREAS Finding shortest routes, truescale distances, and bearings. 
Name  Author, Date  Example, with relevant parameters  Also known as / [Equivalent to]  Properties  EqualArea?  Used by / [Applications]  
Azimuthal Equidistant  said to be used in ancient Egyptian star maps fully developed in 15th and 16th centuries  centered on: north pole, paris, new york  Postel (1581)  Constant radial scale from central point (i.e., all straight lines through the central point are standard lines) Distortionfree only at central point  the Cassinis' floor map at the Paris Observatory (1680s) United Nations Emblem Flat Earth Society [easily computing distances or missile ranges]  
Stereographic  used by Hipparchus, 2ndC BCE renamed “stereographique” by François d'Aiguillon, 1613  centered on north pole  Azimuthal Stereographic [Lambert Conformal Conic with both standard parallels at one pole]  Conformal: local angles are preserved, and local circles are not deformed — at every point east/west scale is the same as north/south scale All circles are preserved Loxodromes (constant bearing / rhumb lines) are logarithmic sprials Distortionfree only at central point Map is infinite in extent  [conformal poles or hemispheres] [oblique versions used to map paths of solar eclipses]  
Gnomonic  used by Thales, 6thC BCE renamed “gnomic” by William Emerson, 1749 renamed “gnomonic” by Augustus DeMorgan, 1836  centered on north pole  Central  All straight lines are great circles Distortionfree only at central point Can only show one hemisphere Map is infinite in extent  [plotting shortest route between two points]  
Mercator  Gerardus Mercator, 1569  Cylindrical Conformal [Lambert Conformal Conic with standard parallels at or equidistant from the equator]  Conformal: local angles are preserved, and local circles are not deformed — at every point east/west scale is the same as north/south scale Distortion is constant along any parallel, and any parallel can be defined as standard Constant bearings (rhumb lines / loxodromes) are straight lines Map extends infinitely North and South  Google Maps (note the dynamic scale bar) [navigating a constant bearing]  
TwoPoint Equidistant (available as a custom projection in ArcMap)  Hans Maurer, 1919 Charles Close, 1921  points at: equator, at 60° E/W, new york / sydney  Distance from any point on the map to either of the two regulating points is accurate (although except for the line connecting the two regulating points, straight lines do not trace great circle paths) Distortionfree only at the regulating points When both points are the same, this projection becomes an Azimuthal Equidistant  National Geographic (for Asia) Ellipsoidal version by Jay Donald used by AT&T for establishing longdistance rates  
Loximuthal  Karl Siemon, 1935 Waldo Tobler, 1966  centered on: london, mexico city, sydney  From the central point (only!), lines of constant bearing (rhumb lines / loxodromes) are straight, and their length is accurate Parallel through central point is standard, with same scale as central meridian; parallels are equally spaced; distortionfree only where standard parallel intersects the central meridian  [determining length of rhumb lines — these (usually) do not trace the shortest distance between two points]  
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REGIONAL AND LOCAL MAPS When the scale used is large enough for overall distortion to be minimal. (Shown with world or state maps.) 
Name  Author, Date  Example, with relevant parameters  Also known as / [Equivalent to]  Properties  EqualArea?  Used by / [Applications]  
Equidistant Cylindrical  attributed to Marinus of Tyre, c.100 CE  show: world, montana  Equirectangular [Equidistant Conic with standard parallels equidistant from the equator]  All merdians are standard, with same scale as the standard parallels No distortion along standard parallels  Mapquest (standard parallels 37.5° N/S) Yahoo Maps (standard parallels 51° N/S)  
Cassini (available as custom projection in ArcMap)  César Cassini, 1745 (for France) Johann von Soldner, c.1810 (for Germany, Great Britain)  show: world, california (using spherical coodinates; ArcMap can only plot ellipsoid ±45° from standard meridian)  CassiniSoldner [transverse Plate Carrée]  Constant scale perpendicularly away from the standard meridian No distortion along standard meridian  British Isles (before c.1920) [regions with predominating North—South extent]  
Mercator  Gerardus Mercator, 1569  show: world, montana  Cylindrical Conformal [Lambert Conformal Conic with standard parallels at or equidistant from the equator]  Conformal: local angles are preserved, and local circles are not deformed — at every point east/west scale is the same as north/south scale Distortion is constant along any parallel, and any parallel can be defined as standard Constant bearings (rhumb lines / loxodromes) are straight lines Map extends infinitely North and South  Google Maps (note the dynamic scale bar) [navigating a constant bearing]  
Transverse Mercator (available in three varieties in ArcMap, all custom projections)  spherical: Johann Lambert, 1772 ellipsoidal: Carl Friedrich Gauss (1822), Louis Krüger (1912)  show: world, california (using spherical coodinates; ArcMap can only plot ellipsoid ±80° from standard meridian)  GaussKrüger (ellipsoidal)  Both the sphere and ellipsoidbased versions are conformal, but with the ellipsoidbased version scale distortion does not increase uniformly away from the standard meridian No distortion along the standard meridian, and at every point east/west scale is the same as north/south scale Map extends infinitely East and West  USGS (with reduced scale factor) [regions with predominating North—South extent] Various US State Plane systems — see reference map  
Oblique Mercator (available as a custom projection in ArcMap)  spherical: Charles Sanders Peirce (1894), Ernst Debes (1895) most common ellipsoidal equations: Martin Hotine, 1946  show: world, alaska panhandle  Rectified Skew Orthomorphic Hotine Oblique Mercator  Hotine’s formulas are conformal, but with an ellipsoid it is impossible to preserve scale along the reference azimuth, and scale does not increase uniformly away from it Scale is (almost) constant along reference azimuth, and at every point east/west scale is the same as north/south scale Map extends infinitely away from reference azimuth  [regions with predominating extent other than North—South or East—West, such as Hawai'i and the Alaska Panhandle]  
Lambert Conformal Conic  Johann Lambert, 1772  show: world, montana  Conformal: local angles are preserved, and local circles are not deformed — at every point east/west scale is the same as north/south scale Distortion is constant along any parallel Map is infinite in extent Special Cases: Mercator results when standard parallels are at or equidistant from the equator. Polar Stereographic results when both standard parallels are at one pole.  The American Oxford Atlas Rand McNally National atlases of Canada and Japan USGS (since 1957) Various US State Plane systems — see reference map  
Polyconic (listed as a “World” projection in ArcMap)  Ferdinand Hassler, c.1820  optimize for: world, north america, california  American Polyconic (to distinguish it from various “modified polyconic” projections) [Bonne with every parallel drawn as a reference parallel]  All parallels are standard and circular, with the same scale as the central meridian No distortion along the central meridian (This projection is essentially obsolete, its major benefit having been ease of construction with numerical tables.)  US Coast and Geodetic Survey, using ellipsoidal form (until c.1920) USGS (until 1957)  
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