The Grammys are the American version of the Brit Awards.
Brits have awards mainly for albums and people: British Male solo artist, British Female solo artist, MasterCard (the sponsors) British Album, British Single, British Breakthrough Act, British Live Act, International Male solo artist, International Female solo artist, International Group, International Album, International Breakthrough Act, Outstanding Contibution of Music.
The Grammys cover loads of stuff: Alternative, Blues, Classical, Children's, Comedy, Composing, Country, Dance, Disco, Film/TV/Media, Folk, Gospel, Jazz, Latin, Musical Show, Musical Video, New Age, Best Album Cover, Polka, Pop, Production, RnB, Rap, Reggae, Rock, Surround Sound, Spoken, Traditional Pop and World. They have a big four of Record of the Year, Album of the Year, Song of the Year and New Artist.
So they're similar, but different.
We also have the BAFTAs, you have the Oscars (for film etc). Above, in the Grammys, you award film music there, we award film music in the BAFTAs (I think, I may be wrong).