Plain English Meaning
A small aluminum cup around the compass magnets that quietly steadies the compass card so it doesn't keep swinging back and forth when the airplane moves.
Full Definition
A non-magnetic aluminum housing surrounding the magnet assembly of a vertical card magnetic compass that uses electromagnetic induction (eddy currents) to slow and steady the movement of the compass card, preventing it from swinging or oscillating excessively in flight.
Context Anchor
Seen in descriptions of how a vertical card magnetic compass steadies its heading indication.
Derivation
The word 'damping' comes from an old sense of 'damp' meaning to stifle or deaden — the same idea as dampening a vibrating string. Here it means slowing or calming the motion of the compass card, not making it wet.