You might think your mom would be thrilled to open a birthday present and find a live rabbit, but actually that's the last thing she's hoping for. Actually is an adverb that means "in reality."
Actually can be used two ways: first, as an adverb meaning "really." You might say, "It's hard to know which of these cookies will taste best without actually sampling them all." The other way to use the word actually is to imply that your expectations have been proven wrong, that the opposite of what you thought is true. You could say, "The hotel room looked spotless, but actually it turned out to have bedbugs."