It's never made clear in the books. It is mentioned, however, that Digory is supposed to a famous professor- so it's likely that his field is pretty well-known. This is further supported that he is supposed to be helping Peter out with some studying in VDT- he's probably a teacher of some fairly common academic subject. 

My guess? Probably something to do with history, philosophy or literature.

It's easy to surmise that his time in Charn and Narnia may have given him a taste for such things. 

Also, it could be assumed that he has the same occupation as C.S. Lewis's tutor, who inspired the character. Does anyone remember what he was a professor of....?

He mentions Plato in The Last Battle so he's probably a professor of philosophy. He also stresses the importance of logic to Peter and Susan, which is arguably part of philosophy. 

Going off of @col-klink's point, it is also possible that the Professor is a proponent of classical education, as grammar/dialectic, logic, and rhetoric are the three essential components (and classical education involves a lot of the Greek philosophers, including Plato). According to a quick web search, classical education began to wane in the early 1900s, seeing a sharp decline by the 1950s and onward. Digory is old enough to have been probably raised with a classical education, but Pevensies may not have at the point of LWW, which might explain his comment about "Logic! What are they teaching in schools these days?"

--Wanderer