Plain English Meaning
RNP is a rule that says how accurately an aircraft must be able to fly a given route or approach. The number tells you how close to the centerline the aircraft has to stay, and the system itself has to watch its own accuracy and warn the pilot if it slips.
Full Definition
A statement of the navigation accuracy, expressed in nautical miles, that an aircraft must achieve to fly within a defined airspace, route, or procedure. An RNP value (for example, RNP 1.0 or RNP 0.3) means the aircraft must remain within that distance of its intended path for 95% of the flight time, and the onboard navigation system must be able to monitor its own and the if the required accuracy cannot be maintained.