Orderly means neat, tidy, and well-organized. An orderly library makes it easy to find the books you want.
A thing or place can be orderly, like a desk, refrigerator, or a hospital, and so can a person, particularly if they are very calm and well-behaved. And when it's a noun, orderly means a hospital worker who does many non-medical jobs and generally keeps things neat and tidy — orderly, in other words. The Latin root is ordinem, "arrangement or row," which originally meant "a row of threads in a loom."