Maps that show a significant proportion of the Earth's surface run into a fundamental problem: the geometry of the surface of a sphere is fundamentally different from that of the plane. For this reason, any planar map of the world must be distorted in some way. This can be seen by comparing a large triangle drawn on the surface of the Earth, which can have three right angles, with one drawn on a plane where the interior angles must sum to exactly 180°. To draw the triangle on a map, either the angles at the vertices must be changed or the lines must be drawn as curves.
A triangle drawn on a sphere, with right angles at all three vertices
There are two primary quantities which can be preserved in a map projection: area, and angle. Area-preservation implies that the ratio of the area of a region on the Earth's surface to area of the corresponding region on the map is constant, wherever that region lies. This is an important property for visualising geographic data without biasing perception towards areas which are artificially enlarged on the map. Angle-preservation, on the other hand, implies that for each small region on the Earth's surface, the shape of that region is preserved in the map with no shear. This is important property for navigational purposes, since compass points on the map remain at 90 degrees to one another, as they are on the surface of the earth. Unfortunately for the purposes of cartography, there is no mapping of the surface of a sphere to (a region of) the plane which preserves both area and angle.
As a result, a large number of map projections have been proposed, each preserving either area or angle, or with other useful properties. The following applet demonstrates a number of projections; click and drag to rotate the globe.