In the Pokémon video games, you are given a starter Pokémon at the beginning who can then battle wild Pokémon. You can catch Pokémon by battling them until they are weak and then throwing a Poké Ball at them. Once the Pokémon is captured in the Poké Ball, you can train it by battling other Pokémon and Pokémon trainers to increase its skills and abilities.
Individual Pokémon have a catch rate that applies to all members of its species. When a Poké Ball is thrown at a wild Pokémon, the game uses that Pokémon's catch rate to determine the probability of catching the Pokémon. To increase your chances of catching a wild Pokémon, you should reduce its health, inflict status ailments and use Poké Balls effective to the type of Pokémon you're trying to catch.
Once you have the Pokémon, you can use it to battle other wild Pokémon and Pokémon trainers. This increases its level and allows it to learn new attacks and skills. When your Pokémon reaches a high enough level, it evolves into a new form, though some Pokémon do not evolve. You can continue to train your Pokémon and capture wild Pokémon until you have caught them all or until you are satisfied with your line up.