Two numbers are called additive inverses if, when you add them, you get 0. For example, 3 and −3 are additive inverses. Every number has exactly one additive inverse. Your teacher may call additive inverses “opposites”; they mean the same thing.
To find the additive inverse of a number, just put a negative sign in front of it, unless it has one already. If it does, then take it away. A special case is 0; the additive inverse of 0 is just 0 again. That's right — 0 is its own additive inverse. This makes sense, though, since 0 + 0 = 0.