In the Pokémon games, the player takes on the role of a young trainer who sets out on a journey to become the strongest trainer in the world. They travel across regions, catching and training Pokémon to battle other trainers and Gym Leaders to earn badges. The final goal is usually to defeat the Elite Four ––the strongest trainers in that particular region––and then the reigning Champion.
The villain's plans differ from game-to-game, but they all share some common elements. The villains usually want to take over or destroy the region, or find some way to control all of the Pokémon in it. In order to do this, they often capture Legendary Pokémon, which are incredibly powerful creatures that can be difficult for even the most experienced trainers to defeat.
One thing that sets apart each villain's plan is their methods. For example, in Generation I's Kanto region, Team Rocket was trying to take over Pokéstops and Gyms so they could control them and use them for their own evil purposes. On the other hand, in Sinnoh's Diamond & Pearl games, Team Galactic was using a machine called the Red Chain in an attempt to control Dialga or Palkia (two of Sinnoh's Legendary Pokémon).
No matter what their methods are, one thing is always clear: The player must stop them!