Let me give a common understanding and then a more sophisticated historical understanding.
The common understanding is that "Son of God" implies his deity—which it does—and that "Son of Man" implies his humanity, which it does too.
He was a son of man, that is, a human being. And he is the Son of God, in that he has always existed as the Eternally Begotten One who comes forth from the Father forever. He always has, and he always will. He is the Second Person of the Trinity with all of the divine nature fully in him.
He is born of a virgin. He had a human father but he didn't have sex with this virgin until Jesus was conceived. He was conceived of the Holy Spirit in the Virgin Mary. Thus he is human—fully human. The Bible wants to emphasize that he is fully human.
So that's the common understanding: he is both divine and he is human—two natures, one person.
The more sophisticated and important historical insight is that the term "Son of Man" doesn't merely align him with humanity. It is probably taken from Daniel 7, And if you read that chapter you'll see that the Son of Man is a very exalted figure: not just a human figure but an exalted figure. It was Jesus' favorite self-designation.
If you do a study of the term "Son of Man" in the Gospels you'll see that he didn't refer to himself most often as Son of God but as Son of Man. He said things like, in Mark 10:45, "The Son of Man came not to be served but to serve and to give his life as a ransom for many." So he calls himself Son of Man very often.
I think the reason he did so is that, on the face of it, Son of Man is an ordinary phrase for "human being." He was born of a man. And there's no offense there: who isn't a son of man? But those with ears to hear could hear Daniel 7, in which he was claiming a very exalted role in the history of redemption. And he meant to do it.
Jesus was very subtle in that he was always opening his identity to those with eyes to see, but he wasn't opening it so blatantly that everybody would come and make him king. He had to steer a very narrow course in disclosing his identity, not just openly saying, "I'm the Messiah, I'm the King of the World. Come and acknowledge me as King." He didn't talk like that.
He was quiet. He was subtle. And he would make claims that were explicit in certain settings and implicit in others. And only when the time was right—mainly when he was on trial for his life, and they said, "Are you the Christ, the Son of the living God?"—did he say, "I am, and you will see the Son of Man coming with great power and glory." So he confessed his open deity right at the point where he knew he would be crucified for it.
So I hope that helps. "Son of Man" has the double meaning of human being and, according to Daniel 7, the exalted heavenly one. And Jesus means to communicate both of those.
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What Did Jesus Mean by "the Son of Man"?
The term the Son of Man occurs in Matthew 32 times, in Mark 15 times, in Luke 26 times, and in John 12 times. In the first three Gospels the title is always recorded as having been used by Christ of Himself and never by angel, by man, or by demon. Of the 12 occasions in John, 10 are from the lips of Christ; only twice was the expression used by men, and then only for criticism and unbelief: "We have heard out of the law that the Christ lives forever: and why do you say, The Son of Man must be lifted up? Who is this Son of Man?" Those are the only two occasions in all the Gospels where the term is found upon the lips of any but Christ. It is Christ's own description of Himself, and it is the term that links Him to humanity and shows His intimate and positive relationship to the human race.
For particular illustration I take the story of the temptation, where the Lord is seen standing entirely upon the level of humanity. He was in the wilderness, being tempted as man, as representative of the human race; and that is not my view merely, it was His own statement. In answer to the first temptation He said: "It is written, Man shall not live by bread alone." That is to say, in effect, I am in this wilderness on the human level, as the Son of Man taking the place every other person has to take. I obey the law of God that conditions the life of humanity.
In answer to the second temptation, He said: "It is written, You shall worship the Lord your God, and Him only shall you serve." Thus, He put Himself within the Divine limitation of every other human life and declared that He was living according to the law which every other human must obey if he would come to the fulfillment of his life. In answer to the third of these temptations, He said: "It is said, You shall not tempt the Lord your God.” Thus, He declared that the law which governed Him was exactly the same as that which governed other people. Therefore, the terms that indicate His relationship to men are those that prove His absolute kinship with the human race, His complete identification with human experience.
Adapted from The Teaching of Christ, Himself, by G. Campbell Morgan.