The main Pokémon games are currently in their 6th generation, with 718 total characters to catch and train (not including alternate forms). Each generation revolves around a different region of the Pokémon world - these being Kanto, Johto, Hoenn, Sinnoh, Unova and Kalos.
In most of the Pokémon games, you are a teenager who receives a starter Pokémon from the local Pokémon Professor. Upon obtaining your first Pokémon, you set off to complete your Pokédex, a device that records the data of Pokémon you encounter and catch.
As your Pokémon battle, they gain experience. Upon gaining enough, they can level up, learn new moves, and have better stats. Certain species of Pokémon can evolve into other Pokémon upon reaching a certain level.
In addition to the main games, there are many "spinoff" games that differ greatly from the main games. For example, the Pokémon Mystery Dungeon series are roguelikes, and Pokémon Trozei is a puzzle game.