The earth is a sphere, so to turn it into a flat map it must be projected an distorted into two dimensions. Map projections can preserve angles, area, or distance, but it's not possible for one projection to preserve all three.
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Equidistant map projections preserve distance preserve distances from one or two points or along a line. The Azimuthal Equidistant projection preserves distances from a central point, meaning that the distance from the central point on the map to some other point is proportional to what it is on the ground. The Sinusoidal projection preserves distances along parallels, so all parallels on the map are proportional in length to what they are on the ground. (The Sinusoidal projection also happens to be an equal-area projection as well.) Note that equidistant does not mean distance are equal between every two point, which is why the distances between Kabul and D.C. differ in these two maps.Click image to Enlarge
On equal-area map projections, areas of the same size on earth are the same size on the map, regardless of where they are located.Click image to Enlarge
Conformal map projections preserve angles. The Mercator projection was often used in nautical navigation because it represents rhumb lines as straight. This means that a boat could pick a heading on the compass and following that same course make it to where they wanted to go. The Stereographic projection is often used in polar regions because parallels become circles and meridians go out as straight lines from the pole. (Note: the above map is not a polar stereographic projection, instead it is centered at 0º, 0º.)
Seeing the above map projections and knowing what they preserve, it is pretty obvious why map projections are important. For example, while the Mercator projection is very good for navigation, it is horrible for comparing area. Notice how Greenland is about the same as Africa, yet in actuality Africa is many times larger (as you can see on the equal-area projections). So, if someone were to be doing some research that involved comparing areas, it is important that they use equal area maps, or all their data would be thrown off.
The Mercator projection used to often be used as the standard world map, which led people to inaccurately believe that Africa was much smaller than it really was. Since countries near the equator (most of Africa) where projected smaller, people thought they were less important. So a few decades ago there was a big push for people to use maps that more accurately represented size. So clearly map projections are important if they can sway how important people think countries are.
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