Hunters - judging is based more on the movement and jumping style of the horse (knees tucked, well footed take offs and landings, easy and steady gaits, etc.) and how well the round goes overall. Your horse needs to be able to keep a consistent pace throughout the course (not rushing fences or hesitating). Hunter courses tend to be very straight forward.
Jumpers - judging is based on how clean and fast the horse and rider can make it around the course without any faults. Style doesn't really play a role in it but your horse needs to be very responsive, balanced, and agile. Jumper courses can get very complicated in order to challenge the horse and rider. Tight turns, different stride counts between jumps, different heights and spreads, and objectives that challenge the rider's ability to plan and execute a course are common.
I recommend you watch one of each show before choosing
For jumpers you need a quick, balanced, and high jumping horse. The jumps tend to be much higher than hunters and have extremely sharp turns making it much more dangerous. 4 faults and you are disqualified and usually each fault is a 10 second penalty. The fastest time wins. The jumps are usually plain too so the horse is less likely to spook. (no bushes or flowers under jumps) but there are liver pools. Must have hunter experience before going into jumpers.
There are 2 types of hunters, equitation and under saddle, and there are different types for each of those.
hunter under saddle is judged on the horses movement in multiple jump courses and a flat class. If your horse isn't a great mover do not expect to pin ribbons. The jump courses are always smooth with no sharp turns. There are no roll-backs or bending lines in these courses. There is no time limit and you will receive a score out of 100. In the flat class you can not sit the canter. (half-seat)
equitation is judged on the riders position and ability to control the horse. The jump courses often have 1 or 2 difficult turns and always 1 roll-back or bending line. Sometimes to show your control you will have to halt after a jump and then continue the course. There is no time limit in hunters and you receive a score out of 100. Usually there is only 1 jump course and 1 flat class. In the flat class you must sit the canter (full-seat).
If you are new to all this I recommend starting with a hack division in hunter under saddle. Then move to equitation and finally jumpers. If you want to go beyond that you can look into eventing.