Pokémon is express as ポケモン in Japanese, where it originated. But, since pretty much nobody east of Yokohama knows how to pronounce ポケモン, we can just call it Pokemon.
Pokemon is a media franchise published by Nintendo, the Japanese video game company owned by The Pokémon Company. Pokemon was created by Satoshi Tajiri in 1996, and it is directed at the creatures called 'Pokemon.'
Trainers have two separate but related goals (in most Pokémon games): to complete the Pokédex (their collection of Pokemon) by collecting all of the available Pokémon species found in the fictional region where that game takes place; and to train a team of Pokémon that they have captured to do their dirty work for them.
If that sounds like a lot of work, well, it is. But, it also is fun. The ultimate goal is to complete the subsidiary tasks and become the strongest Trainer: the Pokémon Master.