Tainted Love/Where did Our Love Go - Soft Cell (Gloria Jones - Tainted Love, Supremes- Where Did Our Love go). The extended version with the Where Did Our Love Go bit is the far superior version.
Gary Jules - Mad World (Tears For Fears original cover featured in Donnie Darko). This is superior to the original in every way, shape and form. A prime example of how powerful a song can be when it's sung right. And that's specific to the vocal delivery, not the lyrics or the "quality" of the voice.
Teardrops - Roosevelt (Womack and Womack sang the orgingal which was a huge hit in Europe in the late 80s). Most on here have probably never heard this song which is sang by Roosevelt (Marius Lauber) a German indie artist. Before each tour he picks an 80s tune and does a rework and a release, with this one being done for his first US headlining tour in late 2016 (following his eponymous debut release which is one of my favorite albums of the last 10 years). One of the keys of his best songs is the integration of killer bass grooves and strong clap beats/percussion and he nails it here. IMHO the original by Womack and Womack is terrible. Unfortunately his following to covers of Fleetwood Mac's "Everywhere" and Fiction Factory's "Feels Like Heaven" haven't been nearly a good.
Some other great covers were already mentioned in here (All Along the Watchtower and Hurt)