What exactly is stochastic calculus?

I’ve read the Wikipedia article on stochastic calculus. Is this commonly taught in university level calculus classes? I took the usual 3 calculus classes at my college back in the late 1980s and don’t remember learning anything like this.

I’m interested in studying this to gain a better knowledge of the financial markets.

No, it’s not a standard part of a calculus course. It’s properly a topic in statistics, and would be taught as such. There are a number of mathematics departments that are offering more and more courses in financial mathematics, though, so if you’re looking for a place to study it shouldn’t be that hard to find.