means The part of a garment that covers the arm, or by extension any tube-like cover that holds or protects something (a record sleeve, a wear sleeve).

from From Old English 'slefe' or 'sliefe,' a word tied to the root meaning 'to slip' or 'glide' — the sleeve being the thing your arm slips into. It's a cousin of the verb 'to slip' itself, and possibly related to Dutch 'sloof' (a covering). So at heart a sleeve is just a slot for slipping into, which explains why a record sleeve, a knife sleeve, and a shirt sleeve all share the name: each is a snug little passage you push something through.