Collecting Pokemon is only half the fun, as anybody who's played the original games can tell you. That "combat power" stat that you see when catching a Pokemon isn't for show - they can fight one another, just like in the original games and the TV show. For now, the only place where that happens is at gyms. Pokemon gyms are a fun place to engage with other players, and can be deep and challenging. When you walk up to a gym, you'll see its color, which is the team that currently owns it, along with the list of Pokemon defending it, and the amount of Prestige the gym has. The lower that Prestige rating is, the closer a gym is to being taken over. We've written here about battles before, so let's dive into how the gyms themselves work.
If a gym is the same color as your team, you have two options when you get to it. You can pick a strong Pokemon to help defend it, or you can spar with the Pokemon that are currently there. Winning those sparring matches nets you some XP, and increases the gym's prestige. Like in a normal battle, attacks will hurt your Pokemon, but when they faint, they'll still have a bit of HP left, so you won't need to use a Revive. If your defending Pokemon gets the stuffing knocked out of them enough times for a rival team to swoop in, however, you will need to use a Revive on them. If you have one or more Pokemon stationed at gyms, you can go to the shop and claim some Pokecoins and other rewards as a defender bonus, but you can only claim it once a day. Keep in mind that defending Pokemon cannot be used in your own personal gym takeover battles until they're out of the gym, whether you replace them or another team takes over the gym.
When you walk into an opposing gym, you can look over the Pokemon to battle, but you can't pick just one to fight - you will always start with the weakest and move to the gym leader, the strongest Pokemon present. When fighting an opposing gym, you can select up to six Pokemon to fight. You can, of course, get with some friends on the same team and mass attack an enemy gym to take it over quickly. That's about all most trainers will need to know about gyms, suffice it to say that the usual tips and cautions apply. Watch your CP, attack types and elements, dodge like crazy, and, of course, don't stand in walkways or other conspicuous areas when waging Pokemon war, so that you don't block off people walking or make yourself an easy target for muggers.