A team-mate is one of the other people on your team. If you play basketball for your school, the other four players on the court with you are your team-mates — and so are the ones sitting on the bench. The word only makes sense from somebody's point of view: you are not 'a team-mate' on your own, you are somebody's team-mate. So people say 'my team-mate', 'his team-mates', 'our new team-mate'. It comes from sport first, and that is still where you hear it most, but it moved into offices long ago. A person you share a work project with is often called a team-mate now, especially by bosses who want the group to feel friendly rather than official — 'talk to your team-mates about it' sounds warmer than 'talk to the other staff'. Notice it says nothing about liking each other. Team-mates can hate each other and still be team-mates; the word is about the shared team, not the friendship. Two spellings are both correct: British writers usually write 'team-mate' or 'team mate', Americans usually write 'teammate' with no gap. Your exam will accept any of them, so pick one and stay with it.