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From Bulbapedia, the community-driven Pokémon encyclopedia.
It is presented as being the officially-sponsored Pokémon League tournament, though unlike what would be thought, not all Pokémon are allowed for use. More specifically, Mewtwo, Mew, Lugia, Ho-Oh and Celebi are all disallowed.
The Cup has a level restriction on it as well, only allowing Pokémon between the levels of 50 and 55 to compete. In addition, the three-Pokémon team chosen by the player at the beginning of a battle may not have a total level sum above 155. This means as well that the six-Pokémon team chosen to compete can't have more than four Pokémon of level 55.
There are four levels that may be challenged; each level up must be unlocked by defeating the eight Trainers on the previous level.