Show jumping, called stadium jumping (or simply jumpers) in the US, is a sport in which horses jump a set course of obstacles in an arena.
Show jumps are designed to be very easy to knock down and the task of horse and rider is to jump a clear round (i.e., get all the way around the course without knocking any of the various poles, plants, or bricks down). Its scored "in reverse" with zero being a clear round and horses losing points or "faults" for knocking a fence down.
In traditional scoring, knocking a fence down is penalized four faults. A refusal is three faults for the first refusal (over the entire course), six for the second, and elimination for the third. A fall of horse or rider used to be penalized eight faults, but in modern scoring generally also results in elimination (due to people trying to carry on riding with injuries). In modern FEI rules, refusals are punished with four faults and the number of refusals for elimination has been dropped from three to two. Time penalties are also awarded for taking longer than the time allowed.
If there are ties, a jump off is performed, which involves a shorter and more difficult course, and the competitors are scored by time. (Speed classes, which are timed from the start, are often seen at shows). In high level competitions, it's not uncommon to do two planned rounds before the jump off.
Other unusual classes include Puissance (a high jump competition, with the highest jump being a wall. The high jump world record was set over a puissance wall, and still stands at 8 ft 1 in) and Gambler's Choice (where jumps are worth points and riders have to jump as many as they can in a set time).
Image source: Ronald C. Yochum Jr. via Wikimedia Commons.