Strength and power are both vital in rock climbing and, despite being related, need to be trained for differently. Strength represents the maximum possible force you can exert. This might be locking off with one hand on a crimp, for example. Power, on the other hand, is defined as energy expended over time. Having lots of power enables you to make big dynamic moves like throws and dynos.
Strength is the most important aspect of climbing. Many people would argue that technique holds that title, but all the technique in the world will not keep you on the wall if you do not have enough strength to hold on first.
Climbing strength can come in many forms. The most important and obvious form is finger strength. A high level of finger strength allows you to grip the wall well and keeps you from falling off. This is often our weakest link in climbing, as fingers tend to tire out before arms. Finger strength can be increased through focused finger exercises like hangboarding.
Arm and back strength allows you to move on the wall. Pulling yourself up and locking off on one arm can require immense strength! Core strength allows you to balance and hold precarious positions on the wall. A strong core will help stop you from swinging and barn-dooring when your footholds are bad. Body strength like this can be built through weight training or ring workouts.
Lastly, leg strength allows you to push yourself up the wall with your feet. Strong legs can take a lot of weight off of your arms and hands, leaving you more energy to climb longer. Additionally, sometimes when there are no finger holds your legs are your only source of propulsion up the wall. You may need to balance on one toe while pushing off that leg. This kind of strength can be built through things like yoga and slacklining.
Power is an explosive force. It is defined as energy expended over time. This means that if you release the same amount of energy in less time you have more power. This can be seen in moves like dynos, where you need to rapidly generate a large force to propel yourself through the air. If you move too slowly you will not gain the momentum needed to complete the move.
Power comes into play in more advanced climbing, where dynamic moves are necessary. These require big throws and difficult catches, so being able to generate a lot of force quickly is paramount. To train for this kind of power, aim for quick movements at full strength. Try doing pull ups where you go up as quickly as possible, and then slowly lower yourself down.
Campus board training is the most famous and effective way to build immense power. Campusing accurately mimics the dynamic movements necessary for climbing. This builds extreme strength and power. However, campus training is very hard on your tendons and must be done with caution. It should only be attempted by advanced climbers looking to further their training.