If you're neat, you're clean and tidy. A neat roommate is usually easier to live with than a sloppy one. And a neat mustache is probably better than a big bushy one.
Your neat handwriting might have made you the pride of your third-grade teacher, and your neat hairstyle will pass muster with your ballet instructor (who insists on a very tidy bun on every dancer's head). You can also use the word neat to mean "great," "cool," or "super." Neat comes from the Old French net, "clear or pure," from the Latin nitidus, "gleaming." Since the 1930s, it's also meant "very good."