The Professor is a kind old man who owns the large house in the country that Peter, Susan, Edmund, and Lucy are sent when they're evacuated from Blitz-besieged London.
The Professor's extremely logical (in terms of the logic in this story) and reminds Peter and Susan not to judge Lucy's story of a magical world called Narnia too harshly, since it's always possible that it is true. When the four children recount their adventures to him, he believes them, and hints that he has personal experience of other worlds, too.
We learn much more about the Professor's background, including that his name is Digory Kirke, in The Magician's Nephew. It's tempting to read the Professor as a stand-in for C.S. Lewis himself. Like Lewis, he's a learned and kind-hearted scholar. Also like Lewis, he values and respects the experience of children more than most adults do.