Nuzlocke - complete the game with a team of Pokemon that haven't fainted. If one faints, it must be boxed or released. You must use the first available Pokemon encountered in an area, and no duplicates. HM slaves are exempt (if your existing party cannot learn certain HM moves)
Wedlocke - exactly like Nuzlocke, but the Pokemon are paired, i.e. married (or civil partnership, if you want). If one of the pair faints, both Pokemon must be boxed or released. HM slaves are exempt. (if your existing party cannot learn certain HM moves)
Monotype - complete the game with one type of Pokemon; if your starter isn't/ doesn't evolve into the type you picked, you can use it until you catch the type you picked. HM slaves are exempt.
Solo - complete the game with one Pokemon. HM slaves are allowed but you cannot battle with them.
Randomiser - use a Pokemon/ number randomiser to pick your team, and use those Pokemon (if available in your game).