Plain English Meaning
It is the chunk of sky around an airport (or group of airports) where the approach controllers are in charge, rather than the en route center controllers. When you are in this airspace, you talk to Approach instead of Center.
Full Definition
A defined volume of airspace, with specified vertical and lateral limits, in which an approach control facility provides air traffic control services to arriving, departing, and transiting IFR aircraft, and to participating VFR aircraft. It typically extends outward and upward from one or more airports and connects with the surrounding en route (Center) airspace.
Context Anchor
Seen in Tower En Route Control discussions, where an instrument flight may stay within one or more approach control instead of entering the larger en route .