I may not be a professor yet, but I do do mathematics, so I think I can answer your question.

Do you walk into an office and start scratching away at a pad of graph paper filled with equations and lines of investigation from the past few months?

You've actually got the spirit of it, more or less. It's uncommon to keep around old calculations though; if you've done it once you can do it again if you need to, but usually you just convince yourself of something and move on.

Sometimes you start with a problem, something you've thought up or not. Your goal is to solve this problem in some satisfactory sense, so you work at trying to understand it. Is it possible to solve it at all? What's stopping it from going through easily? Sometimes the most important question is: why isn't it easy?

You develop a number of properties of the problem and the field(s) of mathematics it sits in. You get a feel for the underlying structure of everything, some kind of idea of the workings of the system. You know that relatively simple rules can describe incredibly complicated behavior--it is the mathematician's job to get a handle on this behavior, to describe it thoroughly and completely. To conquer it. I guess it's a little like figuring out someone else's magic trick.

So you see, mathematics isn't really about computation or calculation. It's about understanding. Sometimes we need to calculate things, sure. But our calculations are not usually the kind of calculation you can do with a computer; they are conceptual and abstract. We'll write down equations and expressions and whatnot, but these are peppered between paragraphs or even pages of exposition.

So yeah, that pretty much sums it up. We look, we explore, we play around with concepts and make them do interesting things. Sometimes it doesn't happen quickly (I'd be embarrassed to say how long I've been working on my current pet problem), and you might not get a damn thing done some days. But eventually you'll get that spark of creativity you needed to look at the thing from a completely original direction, and it's that spark of creativity they pay you for.