Photo from the Leonidio & Kyparissi Climbing Guidebook by Aris Theodoropoulos and Katie Roussos.
Rock climbing is separated into 4 main disciplines.
Bouldering is climbing on low rocks up to more or less 5 meters high with crashpads below to cushion any impact from falls. This discipline expresses mainly the strength of climbers as boulders often consist of few but very hard moves.
Sport/lead climbing is where you climb routes up to 60 meters with a rope. As you climb the rock, you clip the rope into quickdraws (~carabiners) which are clipped onto permanent bolts in the rock. This expresses both the strength and endurance of the climber as routes often consist of many hard moves one after the other all the way to the top. This discipline where I specialise when rock climbing.
Trad climbing is where climbers place their own removable own gear into the rock for protection. This is a very technical climbing discipline and requires a lot of experience and knowledge of how to place gear. Gear is placed/jammed into rock features such as cracks. No bolts are used in trad climbing. However, there are some "mixed" routes where there are bolts, but you must also place your own gear to climb the route safely.
Multi-pitch climbing where climbers use sport or trad climbing techniques to climb longer routes up bigger walls (taller than 60m). A lot of endurance (mental and physical) is required for this type of climbing. On very long routes (can be up to 1000m) it may also be necessary to spend a few nights on the wall.
Competition climbing is separated into 3 main disciplines.
Similarly to rock climbing, one is lead/sport climbing. In a competition, this 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.
Again, as on rock, another discipline is Bouldering. 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.