I'm also a big fan of cover songs. Most royally suck of course, but the ones which manage to successfully translate a song to a new genre succeed in revealing the songs "essence" (yeah, crappy word, but I'm just writing this quickly). Particularly when songs are overproduced, even an alternative genre song which is likewise overproduced can make it clear why a song is just a Good Song.
Off the top of my head, some ones I remember quite liking include:
Johnny Cash's cover of NIN's "Hurt." (agreeing with Jim Henley).
Along with Bruce Baugh, I always thought the Hooters' version of Cyndi Lauper's "Time after Time" was really nice. IIRC they actually wrote the song.
Thumper's ska version of Iron Maiden's "Crazy Train."
Defunct Pittsburgh (then) alt-rock band The Affordable Floors' cover of New Order's "Temptation."
L.L. Cool J's "unplugged" cover of his own song "Momma said Knock You Out." Yes, not quite a cover, but close enough.
Yes's cover of Simon and Garfunkel's "America."
Oddly, or perhaps not oddly, most of the covers I like are done by bands I don't particuarly generally like and are of songs which I didn't necessarily like so much in the original.
Honorable mention to Dread Zeppelin's "Immigrant Song." I'm sure there are more, but I haven't given this much thought.
