I saw a mention for Disturbed’s The Sound of Silence, which is breathtaking, but their first great cover was actually Land of Confusion. Just a badass rendition of an awesome Genesis song.
Also, Guns N‘ Roses have a tendency to do a great job with covers. Their versions of Knockin’ on Heaven’s Door, Live and Let Die, and Sympathy for the Devil are great.
And of course, in my opinion, Metallica are the masters (pun intended) of covers. Am I Evil?, Stone Cold Crazy, So What?, Whiskey in the Jar, and Turn the Page have become classic Metallica songs despite being covers because the band did such an amazing job on them. Turn the Page in particular is as close to being a perfect cover as possible, with the band keeping the mood of the song intact while molding the arrangements to fit their sound and style.