Couple of these people may disagree with, but let's see.
Grace Potter and the Nocturnals- "White Rabbit" (original by Jefferson Airplane)
From the "Alice in Wonderland" soundtrack from a few years ago, this song gives me chills and Grace definitely took this song to new levels.
Johnny Cash- "Hurt" (original by Nine Inch Nails
Two polar opposite people doing this song...there's something about the delivery, plus Cash would pass away shortly after this song came out.
Travis Tritt- "Take It Easy" (original by the Eagles)
Off the "Common Thread- Songs of the Eagles"- country tribute album to the Eagles. Tritt's voice gives it a bit of an oomph that Glenn Frey just misses when he sang the original
Red Hot Chili Peppers- "Higher Ground" (original by Stevie Wonder)
Stevie's version is great, but something about the mix of rock and funk that Anthony Kiedis and the gang add to this song that just makes it even more kick-ass.
It may not be as good as the original but it's still a damn good cover, Faith No More - War Pigs. Love the live version that they do where Mike just doesn't say the words and instead puts random noise in. Great stuff.