From Bulbapedia, the community-driven Pokémon encyclopedia.
Ash's Poké Ball holds a recently caught Pokémon.
Caught Pokémon are Pokémon that have been put in the control of a Pokémon Trainer by the use of a Poké Ball, or by trading with another Trainer. The capturing process typically involves a battle in which the Pokémon is at first weakened and/or affected with a status ailment, at which point the Trainer throws a Poké Ball (or upgraded version) at said Pokémon in order to catch it. Catching Pokémon is central to all canons in the Pokémon franchise; a caught Pokémon can be used in battling and for quests to aid Trainers, whereas Pokémon in the wild cannot, except in some episodes in the anime. Pokémon that are caught, however, do not need to obey their masters; if a Pokémon is traded and the Trainer does not have the badge required, it may disobey and refuse to attack during battles. According to the anime, even with many badges or being under original ownership, many Pokémon still may refuse to battle until trust is gained between the Trainer and Pokémon.
Caught Pokémon are stronger than Pokémon in the wild in the games. At the same level, a caught Pokémon will have more points in every statistic than a wild one; this is due to the fact that caught Pokémon gain EVs from every battle they partake in, and therefore, on training, will gain points in their stats based on what they've encountered. This becomes more and more obvious at the later stages of the game, where the higher levels allow for a greater disparity of stats between caught and wild Pokémon, especially those raised from a low level, giving the Trainer a more decisive edge in wild battles. It's important to note, however, that Pokémon controlled by other Trainers are just as strong statistically (at the same level) as the player's are. Caught Pokémon can level up and learn new moves in battle, while wild Pokémon cannot, and can evolve.
Newly caught Pokémon are no stronger or weaker than their wild counterparts, with the exception of innate IV differences. By using some varieties of Poké Ball, such as the Luxury Ball and Friend Ball, a Pokémon's happiness may be higher than usual upon capture.
A related concept to caught Pokémon is captured Pokémon, from the Ranger series. While catching uses a Poké Ball, capturing requires a Capture Styler, and the captured Pokémon can only be used once for either helping to capture another Pokémon or for its Field Move before it's automatically released.
- In the anime, Ash, May and Max show confusion when they first witness Solana use her Capture Styler. This is because of the difference between the concept of catching a Pokémon and capturing one. The difference in the terms is very slight, but they do refer to separate concepts.