So many songs are covers and most people don't even know it. For example, the song that Zeppelin made popular, Dazed and Confused, was a cover originally written and sung by Jake Holmes and then the Yardbirds. But everyone associates the song with Zeppelin. And then there are songs that have been covered so many times that you don't even know who actually wrote and recorded it first - e.g.: House of The Rising Sun, Man of Constant Sorrow, My Way, etc.
This conversation starts with Johnny Cash's last album - American IV - The Man Comes Around. There are 3 for nomination of greatest covers from that album alone.
Original - Depeche Mode (and Marilyn Manson does a great rendition as well)
Original - U2 (and it was a great song when they did it)
And then of course, there's this. Original - Nine Inch Nails