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 because, 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, exalted heavenly one. And Jesus means to communicate both of those.
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