Rental Pokémon are any Pokémon which a Trainer does not formally own and uses temporarily for a battle or competition. Using a rental Pokémon does not count as capturing a Pokémon for filling up a Pokédex. They are most commonly used to test a Trainer's ability with using any Pokémon, even those they did not train and raise themselves.
In the games
Pokémon Stadium was the first game to make use of rental Pokémon. In this game, the different tournament levels have one of every Pokémon that can enter the tournament available for rental. Each Pokémon has a pre-determined move set for each tournament level. This provides the advantage of making it easier to compete in the different tournaments and to add statues of all Pokémon from the Generation I Pokédex to the Hall of Fame.
Rental Pokémon are used in this game as the same manner as the first Stadium.
Rental Pokémon can also be used in a Battle Now! feature, which semi-randomly generates teams of rental Pokémon for players to use. They are also used in the exams at Pokémon School.
Unlike their Stadium predecessors, these two games do not allow rental Pokémon for the competitions inside the games. However, they still have semi-random rental Pokémon teams available in their Battle Now! modes.
In the Battle Frontier, the Battle Factory is based around rental Pokémon. The player chooses a team of three out of six random rental Pokémon. Noland and Thorton, the Frontier Brains for the Battle Factories, also battle using random teams of rental Pokémon.
Like the Stadium games, rental Pokémon are available. However, not all kinds of Pokémon are available to rent at a time. Rental Passes are also the only kind of Battle Pass available for use at the Gateway Colosseum. Once a Rental Pass is earned, its Pokémon can be swapped between Rental Passes to build a team. Later on players can earn the ability to trade rental Pokémon with their opponents in Gateway Colosseum, and keep them after winning.
In the Pokémon World Tournament, the Rental Tournament and the Rental Master Tournament are, as their names suggest, based around rental Pokémon. The player chooses a team of three out of six random rental Pokémon to compete in these tournaments.
In the anime
Ash and James battled an examiner using teams of three random rental Pokémon in The Ultimate Test. James picked a team of Pikachu, Charizard, and Ivysaur, but was disqualified after he sent out two at once. Ash picked a team of Weezing, Arbok, and Meowth. With this team, he lost to the examiner, but was able to defeat the Team Rocket trio.
In the manga
In the Pokémon Adventures manga
Multiple rental Pokémon appeared at the Battle Frontier in the Emerald chapter. Besides the Battle Factory, rental Pokémon are also used in other facilities (excluding the Battle Dome due to its tournament-style rules) by computer-controlled virtual Trainers working as the challenger's opponents.
During his Battle Factory challenge, Emerald rented a Sceptile, which he noticed to be on a higher level than the other rentals he used, making him realize that it wasn't a rental Pokémon to begin with, and was snuck into the facility by someone. This prompted him to take Sceptile with him. Ruby later revealed to him that Sceptile was the very same Treecko that Emerald had once been supposed to receive from Professor Birch.
All the rental Pokémon from the Battle Factory were later stolen by the villainous Guile Hideout, using them against those opposing him while he was hunting down Jirachi. Once he successfully caught Jirachi, he unleashed the rental Pokémon upon the members of the press present during the Battle Frontier opening ceremony. He also used the rental Pokémon inside the Battle Tower to attack the Pokédex holders until all the rampaging Pokémon were calmed down by Emerald's mud pellets.