The Underdark is a vast subterranean domain which holds thousands of adventure possibilities and myriad threats, including drow, mind flayers, dragons, among others. Because of its nature of being set entirely underground, the Underdark can easily be added to any other world setting.
As of 4th Edition, The layout of the Underdark consists of four basic sections: The Shallows, The Deeps, The Feydark, and The Shadowdark. Each of these sections contains information on its denizens and hazards, and provides three adventures. There are also three campaign arcs, dealing with "aberrant enemies", the Drow, and with Torog (the god of imprisonment and torture, and the lord of the Underdark).
The 3rd Edition Forgotten Realms had their own Underdark with differently named and populated regions. Since most fans learned of the dark elves through R. A. Salvatore's series of Drizzt Do'Urden novels, and since those are set in the Forgotten Realms, that Underdark has become the standard one to many players.
Read more about this topic: Dungeons & Dragons Campaign Settings