Plain English Meaning
Permission from air traffic control to fly a published instrument approach down to the runway. Once you have it, you are cleared to follow the procedure as charted.
Full Definition
Authorization issued by ATC permitting a pilot to conduct an instrument approach procedure. The clearance typically specifies the type of approach to be flown and constitutes approval to descend and execute the published procedure once the aircraft is on a published segment of the approach.
Context Anchor
Heard on the radio during instrument flying when a controller clears an aircraft to begin the final published path toward an airport.