The Elite Four is a Trainer Class and title given to four trainers that qualify as being the best trainers in their respective regions, with only the Pokémon Champion above their skill level. There are currently 7 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 Gold and Silver and 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, VI, and VII, 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 character is defeated at any point, then they must re-battle all members of the Elite Four.
With Ritchie currently being the only exception, members of a region's Elite Four typically only specialize in one specific type of Pokémon (much like gym leaders).
The Kanto, Hoenn, and Unova Elite Four are similar, however, at least one type needs to be changed in order for them to be the same.
The Unova Elite Four is the first one that allows the player to choose the order in which the Elite Four is battled. The Kalos and Alola Elite Four also shares this trait.
The Kalos Elite Four are the first Elite Four who's Pokémon do not go up levels when the player returns to the Pokémon league for their first rematch, in fact they stay at the same levels as the first time the player fought them for all rematches.