A category is a group of things that share some commonality. Think of the game show "Jeopardy!," and how each column is a different category of questions — like "Literary Characters" or "Potent Potables."
The noun category has the sense of ordering or sorting. In Biology, the science of taxonomy is all about categorizing all living organisms. It might be an easier task to categorize books in a library according to genre (e.g., fiction, non-fiction, mysteries, etc.) or to categorize the shoes in your closet according to season.