Secular philosophers and scientists hold to a general consensus that the mind is that part of the brain that deals with self-awareness and self-purpose. Brain is matter (e.g. chemical activity) and the mind is self-awareness (e.g. the experience of living.) It should be noted before moving forward with this post that there is broad disagreement however as to how the mind and the brain interact. Are they separate of one in the same? This is material for another post (I suppose.)
Biblically the mind is much more than the secular understanding. In the following scripture passages from the English Standard Version the mind is the initiator of all moral action. "With our minds we are responsible before God, and we respond to him, either for or against (emphasis mine)." (Dr. Edward T. Welch, Blame It On The Brain, 1998, page 36)
"Do not conform any longer to the pattern of this world, but be transformed by the renewing of your mind. Then you will be able to test and approve what God's will is—his good, pleasing and perfect will." (Romans 12:2)These passages show how the word heart is a useful synonym for mind. However, it may not be so clear, but biblically mind is not synonymous with intellectual activities. Biblically, because they are not moral functions, intellectual activities like memory, logic and academic abilities are more related to the actual physical brain. I think this is why the bible refers more to the mind as the heart, or the spirit, and even inner man or inner self. The big distinction between secular and biblical views of the mind come down to the moral aspect not to the intellect.
"having the eyes of your hearts enlightened, that you may know what is the hope to which he has called you, what are the riches of his glorious inheritance in the saints," (Ephesians 1:18)
"They are darkened in their understanding, alienated from the life of God because of the ignorance that is in them, due to their hardness of heart." (Ephesians 4:18)
" And you, who once were alienated and hostile in mind, doing evil deeds," (Colossians 1:21)
To summarize, biblically, the immaterial substance (the spirit) of man is the moral acting side of man. When I think of Spirit I sometimes get stuck on thinking it is only the non-physical side of things, but what is really important to grasp is the Spirit means mind, heart, soul and these are all one in the same, in the sense that it is through them they we either move toward the Creator or away from Him.