A map projection is a transformation of the locations on a sphere or ellipsoid into locations on a plane. The mathematician Carl Friedrich Gauss proved in the nineteenth century that a sphere’s surface cannot be projected onto a flat plane without distortion. The same holds true for other curved surfaces like ellipsoids. So any map projection introduces distortion. Different map projections have different distortion properties.
A map can’t preserve all of the metric properties such as angles, areas and distance. Thus map projections are classified by the metric properties they preserve.
Introduced in the sixteenth century the Mercator Projection is still in widespread use today. As a conformal projection it preserves the look of smaller shapes. Lines of constant bearing are straight on Mercator maps, so they are used as navigation aids. The big downside of Mercator is that areas are inflating with latitude. So on a Mercator map Greenland looks nearly as big as Africa, while in reality Africa has about 14x the size of Greenland. A simplified variant of Mercator (called Web Mercator) introduced by Google is the de facto standard for web mapping, being used by Google Maps, Open Street Maps or Bing Maps.
The Miller projection is a modified Mercator projection. The inflation of areas near the poles is reduced, so Miller maps look rather similar to Mercator maps without the extreme size distortions in northern and southern regions. Because of these compromises Mapswire.com uses Miller projection as its standard projection for most of its maps.
The Robinson projection is another compromise projection. Size distortions are further reduced at the cost of increased shape distortions.
The Mollweide projection is an equal-area map projection. It preserves the size of figures, but heavily distorts the shapes when getting nearer to the edge of the map. Mollweide maps are especially used for global maps where its equal-area property helps to display global distributions.
Looking somewhat similar to the Mollweide projection, the Winkel Tripel projection is a compromise projection reducing trying to reduce different distortions and works very well for world maps. In 1998 the National Geographic Society started to use Winkel Tripel as its standard projection for global maps.
This conformal projection is often used for medium scale maps as countries or continents. It doesn’t work well for world maps.