Plain English Meaning
An SIAP is a fixed, written-down recipe for flying down through cloud or low visibility to a runway safely. The pilot follows a published chart that tells them exactly where to fly, how low they can go, and what to do if they cannot see the runway when they get to the bottom.
Full Definition
Standard instrument approach procedures are published, FAA-approved step-by-step procedures that guide an aircraft from the en route environment down to a position from which the pilot can either land or execute a missed approach, using only cockpit instruments for navigation. Each SIAP specifies the navigation aids or satellite-based guidance used, the and to fly, the that must be maintained at each , the visibility required to land, and the if the is not seen.