Competition climbing is separated into 3 main disciplines.
One is lead/sport climbing, which is basically who can get the highest up a route tied on a rope and clipping the rope through specific anchor points on the route. The routes are different in all competitions, and the main focus is on how long each climber can last on a sequence of many demanding moves, mentally and physically. There is usually a time limit of 5-7 minutes, and once the time runs out, the climber's attempt is finished. The speed of each ascent only counts in the case of a draw.
Another discipline is Bouldering where climbers compete on walls of around 5 meters high with a crash pad below. There are usually 4 or 5 boulders in each round of a competition, and the climbers get 4 or 5 minutes to try the route as many times as they want in an attempt to top the boulder. The number of attempts counts only if there is a draw in the number of tops the competitors get. Other than the top, there is another hold on the boulder called the "bonus" which still gives you a score if you don't top the boulder.
The last discipline is Speed climbing where it's basically a head to head battle where the climber's race to get to the top of a 15 meter route as fast as possible. Unlike Lead climbing and Bouldering, the route is fixed, so competitors from all around the world can train and perfect their movements on the route to be ready for the competitions.