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 catching process typically involves a battle in which the Pokémon is at first weakened and/or affected with a status condition, 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. 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. 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. 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.
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 friendship may be higher than usual upon being caught.