Pickups come in a couple types, active and passive. Some major companies to get your research started are Dimarzio, Seymour Duncan, and EMG. Pickups also come as humbuckers (the blocky things like on a Les Paul) or single coil (the thin ones like on your typical fender).

Pickups can be swapped around between guitars if they fit the cavity, assuming there's nothing funky with the wiring. It takes a bit of soldering, it isn't the type of thing you can do on a whim with a screwdriver and set of pliers.

Some smaller but high quality pickup companies, just for your entertainment, are Kinman, Rio Grande, Bare Knuckle, and DeTemple.

Hope that helps. Start doing searches, there's probably dozens of topics already here on every aspect of pickups.