A song that is a cover by another artist. I think this has to be Tori Amos' cover of I don't like Mondays, originally by the Boomtown Rats.
Tori Amos was I think the first musician I really got intensely into, beyond just enjoying the sound of somebody's music. The single Cornflake girl was on the radio a lot in the mid 90s, and I quite liked it but didn't have any context. Then I met MK when we were both up for Oxford interview, and became instant friends. He put a lot of effort into supporting me through a somewhat bumpy transition from sheltered child to independent person, including dealing with a bereavement that hit me really hard when I was 19. He's also responsible for introducing me to digital socializing (email, instant messenger, Usenet to an extent, and the wonderful world of peer-to-peer file sharing). And he played lots of Tori songs for me when I was sitting in the dark crying about letting go of childhood naive optimism. I bought Little earthquakes on CD, and had access to a lot of Tori's oeuvre for all of the 90s via not entirely licit digital copies. Not only Tori Amos, there was a lot of alt stuff especially goth that I picked up from doseybat, but Tori Amos was pretty much the soundtrack of inventing myself as an adult.
I don't like Mondays was almost a novelty thing in a way, recorded with a bunch of much less successful covers, of things like Smells like teen spirit which really doesn't work for Amos' musical style, most of which were never commercially released. This one did make it to Strange little girls, the concept album of gender-bent cover songs, which I was never fully convinced by. I haven't been strongly into Tori Amos' music since 2000, not that I think it's bad but it isn't part of my psyche in the way that the 90s material is. But anyway, it's a remix of a song written in response to a school shooting in the late 70s. The original is meant to be ironic, but it comes across as so inappropriately jolly that it often gets played on the radio as a joke song, here's one to cheer you up from your Monday commuting blues... Tori Amos' cover is a total reworking, without any irony at all, just sadness about a teenaged girl turning a gun on her schoolmates.
So it kind of epitomizes why Tori Amos meant a lot to me at that time in my life; she wrote and performed beautiful songs (she's a classically trained musician) about serious subjects which she took seriously. But that seriousness isn't about glorying in the violence and ugliness, it's about challenging it. ( video embed, audio only )
As a bonus, have kd lang's cover of Leonard Cohen's Hallelujah. It's a song that gets covered way too often, nearly always as a kind of soppy lovesong that really fails to do justice to the extremely powerful original. So basically I hate Hallelujah covers, except this one. Again, it's very different from Cohen's original, but it's an emotionally serious interpretation in its own right which doesn't cheapen its source material.