Biting signifies an aggressive attack against a particular idea, concept, or position.
Your subconscious is trying to warn you of your vulnerability physically or mentally (insecurities).
Who or what is biting?
If it is a person who bites, on a deep level you believe that, that part of yourself that is like the person is could be sabotaging you. Self sabotage.
If it is an animal or machine it represents a person in your life that has the same characteristics as the animal or machine and you feel deeply inside that they are attacking you. This could be true or just an overblown fear. Feelings and reality are not the same.
What is being bitten? Perhaps in real life you could concentrate on strengthening your security or confidence in the area of wherever is being attacked to avoid this dream from reoccurring. Look up the significance of the body part or thing that is being bitten.
Is it playful? There is no real threat, but aggressively destructive no less. This is a fear you have but you are aware enough to know that it will not harm you.
Is it menacing or terrifying? Your insecurities may be out of control. Overblown in your own mind. You may need to achieve some kind of balance between reality and fear.
Does it actually make contact and leaves a wound or tear? You are internally wounded by your fears perhaps by some past experience that makes you fearful and an un-whole person, damaged or scared to simply just be who you are.
Does it threaten to bite, but doesn't? There is no real danger, just fear. You are actually quite safe, but your mind it determining a bogus threat.