It means "very small" - almost meaning the smallest possible - like tiny or minuscule. In this context, it's pronounced "my-noot" and not "min-ut" (like the time). He had to describe the situation in minute detail. The party planner paid attention to the most minute details. I'm struggling to think of practical examples where it's used apart from "details." The two words very often go together in usage.

‘minute attention to detail’ means taking into account even the smallest details. These forms are very complicated and will be rejected if there is a single error. Minute attention to detail is required when filling them out. The director is known for his minute attention to detail, but surprisingly this film contains an anachronism.