The Pokedex is an encyclopedic device present in all of the main Pokemon games and TV shows. It was created by Professor Oak of the original games and TV series and it documents every Pokemon that the player (or Ash in the TV show) encounters.
In the TV Show, as well as some of the games, the conceit is that players (and Ash) will be going out into the world in the attempt to capture and document new Pokemon that haven't been seen before. In the games, the Pokedex will register a Pokemon that has been seen by the player, but it will not complete its entry on the Pokemon until the player has captured one (implying that during down time the player inputs the information).
What, specifically, the Pokedex looks like and what information can be found in it varies from game to game. In-game, the different countries or "regions" of the Pokemon World all have their own specific Pokedexes due to the different pokemon types that are common in each region. In some games regions can be unlocked once the player has completed significant parts of the game. In Pokemon Gold & Silver, for example, once the player has completed all the main tasks in the Jhoto region, s/he can unlock the Kanto Pokedex and travel to Kanto to complete the Kanto Pokedex.
Each Pokedex entry comes with various information regarding the selected pokemon including things like height, weight, the area the species can be found, "cry" which plays an audio clip of the noise it makes, and a sentence or two about the pokemon's species.
At its best, the Pokedex helps develop world building of the Pokemon universe. At its worst, it introduces plot holes, contradictions, nonsense, and pants-shitting horror.