If someone is biting you, the dream may be an expression of hostility that you are intuitively picking up on. Is there someone around you who may be threatened by you, jealous of you, or angry at you? As in the dreams of you biting someone else, looking at who is biting you is a big clue in understanding what the dream is trying to communicate.
Being bitten by insects or animals can represent fears, nuisances, or other things that are "gnawing" at you.