If a
projection is the
transformation of
points from one
space to another, cartographic projection is the particular case of mapping from a
spherical surface onto a
plane.
Collectively the mesh of lines of latitude and longitude on a sphere are called a graticule. If a single graticule line can be transferred to a plane, the other lines can be arranged about it in a mathematically orderly way; not the same as on the sphere but related. These arrangements are map projections, on which lines of longitude are called meridians, and of latitude, parallels.
Such projections can preserve directions or distances but not both, except from a single point. Those that do preserve bearings and distances from a single point are called equidistant; those that preserve bearings from any point are called conformal. Maps that preserve area are called equivalent or equal-area, but they give up both equidistance and conformality.
No map can possess more than one of equidistance, equivalence or conformality, and many maps have none.
This means that if your map shows the proper sizes of countries, it will be unreliable for compass navigation. That's one reason why, historically, the Mercator projection has been popular: it's conformal, meaning that bearings are correct, though distances and areas are not. Greenland is *not* that big. Another reason is that on the Mercator projection, the meridians are parallel, making lines of constant bearing (rhumb lines) turn out straight. It's also one of the few that maps to a rectangle, which has been more important than you might think in making it the standard in books and atlases.
For statistical or political purposes it's useful to have maps that show equal area, such as the Mollweide projection, which has a central meridian perpendicular to the equator and the other parallels. Its defining feature is that its 90° East and West meridians together form a circle. The sinusoidal, or Sanson-Flansteed, projection is another equivalent projection.
The stereographic projection may be familiar from the National Geographic Society's logo, where the parallels diverge as they get further from the central meridian. It also is conformal for navigation, and has the added virtue over the Mercator that both oblique and great circles on the globe are circles on the map as well.
I'll also mention the orthographic projection not only because it's what you'd see from space, but because it's what astronomers use for star charts and maps of the Moon's surface.
All the above are continuous projections. If you're willing to interrupt the map into gores like a peeled orange and lose almost all distance information, you can get closer to having both conformality and equivalence simultaneously.