One of my favourite covers is Adele's version of "Make you feel my love", by Bob Dylan. The original one is really good, but Adele's voice is just amazing and really suitable for this song.
One of the best covers ever in my book is Disturbed's Land of Confusion over the original by Genesis
Disturbed also covered Tears for Fear's "Shout" (Shout 2000- The Sickness) and Judas Priest's "Living After Midnight" (Living After Midnight- The Lost Children). They're real good at remaking songs better than the original. On a separate note, RHCP's "Roller-coaster" (original by the Ohio Players) is like ten times better than the original also.