From Bulbapedia, the community-driven Pokémon encyclopedia.
A trainer card in the Pokémon games is the card that displays much of the information about a trainer. They seem to have appeared in all four generations of Pokémon games, however, they are only officially known as trainer cards in Generation III and Generation IV.
Several things are shared among the trainer cards, including displaying the trainer's name, their trainer ID number, how much money they have on them, how many Pokémon they have in their Pokédex, and the amount of time they have been playing their game.
In Generation III, a back side was introduced that displayed such data as how many times a player had played in link battles, won and lost, traded, and even how long they had been playing when they first defeated the Elite Four and Champion to enter the Hall of Fame. In Pokémon FireRed and LeafGreen, a trainer's party could be photographed in the Rocket Game Corner in Celadon City and put on the back as well.
When Generation IV was released, the ability to keep track of the precise date on which the game was begun and the date and time at which the Elite Four were defeated was added. An area for the trainer's signature was also included. In addition, when communicating in the union room, other players can ask to view the players trainer card, which has a picture of a random trainer class, which can be changed by talking to a woman in the Oreburgh City pokemon center.
In Generation I, II, and III, the badges that had been earned were displayed on the bottom of the card, however, with the bottom screen of the Nintendo DS adding to the display area in Generation IV, an actual badge case was added, where trainers could actually shine their badges using the stylus.