This exceedingly simple projection preserves the area of
continents. That is the most important property when glancing at a map of the
whole Earth. The first instinct we have is to compare the size of geographic
features of the planet. While things are squeezed and amorph near the poles,
shapes are accurate near the Equator.

Mercator

This one preserves angles, which means that shapes are
accurate. However, everything near the poles is exceedingly
inflated. So much so that the poles are actually infinite and
cannot be represented.

Equirectangular

Notice how meridians (vertical) and parallels (horizontal)
form squares? This projection maintains heights. A distance
measured vertically corresponds to the same number of kilometers everywhere on
the map. Of course, a horizontal distance is only the same along the parallel:
as the ruler goes closer to the poles, that distance represents a smaller and
smaller amount of kilometers. (Also, this is a good middle ground of pole
squishiness between Lambert Cylindrical and Mercator.)

Interrupted Sinusoidal

This one is a complement to the Equirectangular. It maintains
widths. A horizontal ruler measuring a distance will
correspond to the same number of kilometers throughout the map! Since the
default rendering of this projection makes the map converge to a single dot at
the poles, I favour this interrupted projection (or "gore map"),
which does not distort shapes quite as much. As a bonus, this projection also
preserves areas, although it is arguably not as readable as
the Lambert Cylindrical.

Winkel Tripel

We have a lot of maps with many incompatible properties. This projection gives
an approximation of all desirable properties in a map. Hence, the name, German
for "triple": it tries to preserve
areas, angles and distances.