A geographic projection is a specific way of shaping a map so it can be displayed.
The word ‘projection’ is sometimes used to describe both the display of data and the way it is stored - what we often call the datum instead. Here we’re talking about projections in terms of display only.
Since the world is similar to a bumpy ball and both paper and computer monitors are flat, projections need to warp, bend, and cut the Earth’s surface to fit. When they change the world in these ways, they make different compromises - one projection might make a country look smaller than it is really, relative to other countries. Or a projection will show a line as curved even though it would be straight if you looked at it from a perspective on Earth.
The important thing to remember about projections is that every choice is a compromise, and there are many choices. There are many projections to choose from, with years of history and many specific uses. Some projections are created only to show one specific country, or optimized to fit a rectangle, square, or circle shape in a design.