I think it's officially the "African Cup of Nations". Yes, it is a little unusual; "The African Nations Cup" makes much more sense. Perhaps they thought it would be awkward because of the initials ("ANC").
I didn't see any of the games myself – they only seemed to be on tv here in the UK at strange times when I couldn't watch it. But we have a lot of African players over here in the Premier League anyway, so we get to see a fair bit of African football!