Anybody = one of many different people. "Is there anybody here who can help me?" This is asking for one, unspecified person. Anybody could also mean several people (but not SPECIFIC people) out of many: "Anybody want to come with me to the bar?" This is asking about multiple unknown people.
Nobody = not even one of many different people. "Nobody is here to help me!" "Nobody wants to come to the bar with me."
Sometimes they are used in a very similar way: "There isn't anybody" is the same as "There is nobody"