With the exception of the .50 BMG, there are four basic types of reloading setup. (The .50, I believe, needs an extra-large one.)
The Lee Loader comes in a box not much bigger than a couple of cigarette packages. Instead of a press, it uses a hammer and your effort to cycle the rounds through. It turns out perfectly good ammo, but takes longer, is noisy and takes more effort. Each Lee Loader is for one specific calibre.
The next three all use interchangeable dies, one to swage the case back to the proper size after having expanded in the chamber, one to kick out the primer and bell the case mouth just a bit and the third to push in the bullet and crimp it in place. (There are variations on that theme, but it's a good intro.) The difference is in the press, the gizmo that uses mechanical leverage instead of a hammer. The thing is, all you have to do is change the dies (each set calibre-specific) and a couple of other minor things and you can reload anything from .17 to .458.
The single-stage press uses one die at a time. You have to run a bunch of cases through, swap out that die for the next, run them all through, swap out the die... It's slower than the other two, but is simple and cheaper. It's where most people start and most never see the need to move past it.
The turret press is much the same, but all the dies are mounted in a moveable plate, allowing you to switch dies much faster. They're a bit more expensive.
The progressive press is like a miniature factory. Every time you pull the handle, components feed in from hoppers and a finished round kicks out. For people doing beaucoup shooting, they're the way to go, but they're considerably more expensive and they are more complex.
I would strongly recommend you look at a book called The ABCs of Reloading. It's one of the best introductions to reloading and will answer most of your questions.