Geographic projections are mathematical procedures that display the objects on the surface of the earth on a two-dimensional surface. If you imagine trying to cut and flatten out a basketball into a perfectly flat surface you may see why this is not trivial. Geographic projections create two-dimensional maps by tracing vectors from an originating point through the feature on the surface of the earth to a shape wrapped around the earth. The map is then the image after “unrolling” the shape placed mathematically around the earth. For example, a Mercator projection uses a hollow cylinder that is tangential to the equator. Imagine a light source in the center of the earth that shines through the earth and projects the features onto the tube. The image on the unwrapped cylinder is a Mercator projection map. In the image, the features close to the equator are well represented. However, features that are further from the tangential line become distorted. A projection cannot preserve all the of the quantities: distance, shape, directions, and area. Some of these will be distorted with any projection.
The transverse Mercator projection (below) is similar except that the hollow tube is now horizontal so that the tangential line is north—south. The universal transverse Mercator, or UTM, projection uses this setup and rotates the cylinder to be tangential to many lines of longitude around the globe, resulting in many UTM zones which are vertical slices with the tangential line running through the middle of each. This results in a good representation of features along the midline and fairly good representation of most quantities within the zones because these are relatively narrow swaths.
Yet another type of projection is a conic projection (below). This projects from the datum to the surface of a cone. The cone may be tangential at one line of latitude or intersect the globe along two lines of latitude. The resulting map is quite good at preserving quantities around the parallels of intersection if these are close together. A good rule of thumb is to have one intersect 1/3 of the way up the map extent and the other intersect 2/3 of the way up the map. This projection type is valuable for large regions that are wider than tall.
Different projections preserve different quantities, so it is important to know what values you need and to use the appropriate projection. There are many variations on these to optimize the representation of local areas resulting in thousands of different projections that are unique combinations of datum, estimated spheroid shape, grid systems, and projection types. In all three examples above I placed the datum in the center of the earth, but this is not a requirement. It simply made the rendered animation easy to code.