What would happen if one of the layers of the earth vanished?

The Earth can't explode. Even if we blew up all the atomic weapons ever created, it wouldn't even put a dent in the earth. If you ignore the impossibility of it actually happening, and assume the earth exploded anyway, this is what would happen:

The Moon and the dust left over from the Earth's explosion would continue on their orbit around the sun. If the explosion wasn't too strong, gravity would pull the dust back together to form a new planet. However if the explosion was strong enough, the fragments of the earth would be scattered to form a new asteroid belt, right where the Earth's orbit was.