I've heard of a catch rider, but have never used one or been one myself. Typically you have to already have established yourself as a high level rider for people to want to pay you to show their horse.
A catch rider is someone who gets hired to ride a horse at a show. It normally involves the warm-up before hand. Normally a catch rider is well known or well established in the area. They must be easily adaptable to a variety of horses (greenies to well schooled). If a CR doesnt follow what the trainer wants them to do in the ring, they may not get asked to ride again.
A lot of jockeys and drivers (for Standardbreds) are known as "catch riders/drivers" as they follow the circuit. Some barns have a rider/driver that gets put down on every horse that comes out of it, but a lot of other horses get ridden/driven by who is available when you enter the horse in a race.
Prob with catch riding though... is if you do anything with USEF and you're riding as an armature, and they find out you've been paid to ride someone's horse for them, you're supposed to be registered as a pro.... you will lose any winnings as a result.