How much math does one need to know to understand discrete math/structures for computer science?
Normally, classes at universities have prerequisite lists. If there are some courses on the list you haven't taken, you should ask the professor whether you really need them. 

Discrete math courses can vary dramatically in what you really need to understand them. You may or may not need to have done proofs; (some discrete math classes teach you how to do proofs). I'd guess you probably don't need to know calculus. Calculus isn't really needed to understand discrete math, but if calculus is a prerequisite for the class, there are a number of good examples and homework problems that the professor might use that would indeed require calculus. And you can certainly teach discrete math classes that require basic abstract algebra as a prerequisite.