Display image based on category

I want to display an image, based on category. I’ve tried looking through the codex but I’m going cross-eyed trying to figure this out. And yes, I’m aware that having multiple categories makes this a bit messy, but right now, I’d just like to get the first step down, based on a single category that an entry is in. But I can’t figure out how to retrieve the category for entries. What to do?