See an informative animation and further explanation. Both are being pulled by the earth due to the gravitational force. The moon stays in orbit due to it having the appropriate tangential velocity that keeps it from coming closer to the earth's surface. The moon, however, is still accelerating at the rate any object would have at that distance from the earth. In a sense, the moon is falling around the earth rather than into the earth.
True weightlessness occurs when there is an absence of gravitational force on an object.
This is not a likely occurence since every mass in the universe is attracted to every other mass in the universe with some force of gravity. Apparent weightlessness is the sensation of not experiencing any external contact forces. This occurs when an object is in free-fall. Even though the object may have a large force on it, it wouldn't register on a scale that was in free-fall with it. Astronauts appear this way on the space shuttle.