I know of two.
The common plastic and the Kadee coupler. The plastic is low end and they should be compatible with themselves. The Kadee has a wire for remote uncoupling of the knuckle. Other brand exist but I would think that they would be compatible with either of these.
Most off the shelf equipment comes with Rapido couplers, those G shaped thingies. More and more of the higher end stuff is coming with Kadee compatible knuckle type couplers so you're going to have to watch out for that.
Couplers... what a mess!. This is posted in N scale so let's talk focus on that. There are two basic styles of N scale couplers: rapido and knuckle. Rapido couplers are sort of square hooks and don't look realistic at all. They also cannot be decoupled remotely. Inexpensive stuff from China usually comes with rapido couplers.
Kadee introduced the first knuckle couplers, but they split into two companies. Kadee kept all products in HO and larger scales and Micro-Trains Line got all the N scale and smaller products.
Knuckle couplers are intended to look more prototypical. Most N scale knuckle couplers are compatible, but not all. Micro-Trains Line (MTL) and Unimate are two popular brands. Some knuckle couplers have metal "hoses" hanging down from the couplers in an arc. These are for decoupling remotely. This is accomplished by parking the couplers over a magnet. It takes some practice.
Some rivet counters seem to think that standard N scale knuckle couplers are too large to be prototypical and they like to use MTL Z scale knuckle couplers to replace their N scale ones.
A variation on the knuckle coupler is the new Kato "kinematic" coupler that closes the gap between passenger cars when running on straight track but allows the gap to expand on a curve. Check out the Kinematic coupler video. These couplers only work with Kato products designed to accept them.
If you have locos and cars with older rapido couplers they can be upgraded to knuckle couplers. Many hobby stores carry replacement couplers and some will do the replacement for you.
That's only N scale, and in the United States. Every scale has different coupler types and there are even some European couplers types not found on American prototype trains.