Elite Four is a Trainer Class and title given to four trainers that qualify as being the best trainers in their respective region(s), with only the Pokémon Champion above their skill level. There are currently 6 sets of Elite Four members and 1 champion, corresponding to the number of Generations that the main series games have been through (although Kanto and Johto share their respective Elite Four location, the Indigo Plateau which technically is in the Kanto region).
In order to challenge the Elite Four, a trainer must first acquire a total of 8 Badges (16 in Pokémon HeartGold and SoulSilver, but you first fight them with 8). Upon battling them, the challenger must face all of them consecutively. In Generations I-IV, the members of the Elite Four are battled in a set order. In Gen V and VI, they can be fought in any order. Once defeated, a trainer can battle the regional Pokémon Champion for their title. However, in every region, if the player is defeated at any point, then he/she must re-battle all members of the Elite Four.