Fins do a lot! Fin design can get pretty complex pretty quickly, but they are essential for getting the most out of your board. Without fins you would be paddling in circles. Most boards come with fins already, so for the most part, people are looking for a replacement fin, or an upgrade.
Different disciplines and style of boards have very different fin shapes. Surf fins are shorter and stubbier with swept back shapes. All round fins are taller versions of the surf shapes. These are designed to help you turn. Touring and race fins are wider at the bottom, and extend deeper into the water. They are designed to keep you going in a straight line.
Another key thing to note on fins, is what fin box they are designed to fit into. The most common type on SUPs are the US Box. This is what most race, touring and inflatables use.