The mind is is responsible for your consciousness, subconsciousness and unconsciousness; it perceives things, deals with emotions, feelings, reasoning, logic... it enables you to make decisions on what your physical body responds to, etc... I don't think there is a biological definition of the mind, since it doesn't exactly exist in the physical world... Correct me if I'm wrong.
The brain is your physical, concrete, biological organ located between your ears. =)
Simplistically speaking, the mind is the abstract, intangible counterpart of the physical brain.