A straight shot squeegee folds down or flares out at the end, like the tip of an arrow. These are designed to get on the other side of the paint from a broken ball so you can pull it out in one motion. These tend to be a quick and dirty solution designed for speed, not to necessarily clean all the paint from your barrel, and because they're long and rigid they can difficult to carry in your gear. These work best with an elastic lanyard.
Pull-through squeegees are rubbery, so they are cheap, and easy to store. However, to use a pull-through you first have to remove the barrel completely. Since they don't maintain their shape, they can become tangled and have to be straightened out before use. Like your standard straight-shot, they don't get all of the paint clear of your barrel. Some pull-through squeegees have a disc on them, so they function similar to a straight shot.
Fluffy squeegees, also sometimes called foldable squeegees because most have a bendable middle section, are designed to swab out a barrel completely on the field. Many varieties come with a swab on one end and a disc on the other end, giving you the functionality of both a fluffy squeegee and a pull-through, like the Barrel Maid.
There is one other specialty squeegee, used to clean breaks and leaks out of pods.
Squeegees are usually best thought of as disposable. They will only soak up so much paint before they need to be replaced. So it's a good idea to carry additional swabs, just in case one is tapped out, so you aren't left high and dry in the middle of a close game.