In a dream in which we are biting a person or an object, we have returned to very basic drives of anger and aggression, and behind that is the need to protect ourselves or our personal space. Being bitten by a snake often represents some kind of initiation.
Psychological / emotional perspective: To be biting into something such as fruit within a dream indicates that there is literally an idea or a concept which we need to get our teeth into.
Material aspects: Being bitten in a dream may show that we are experiencing aggression from someone else, or conversely that our own aggressive instincts in our waking lives are not under control. We are feeling threatened.