RNAV means Area Navigation, which describes ways of flying directly across country without doglegging all over the place, or having to pass over radio fixes, which saves fuel and makes better use of airspace.
There are nearly as many RNAV standards as there are varieties of airspace. For example, B- and P-RNAV are used in Europe, MNPS over the North Atlantic and RNAV 1 and 2 in the USA (RNAV 1 = P-RNAV).
Performance Based Navigation is an ICAO attempt to provide standardisation, where the emphasis is on specifications rather than equipment.
RNP (Required Navigation Performance) is a measure of the navigation performance needed to operate within certain airspaces. In common usage, this means the lateral accuracy in nautical miles that an aircraft is expected to maintain for 95% of the time, relative to a desired flight path.
The system accuracy takes after the specification - for example, RNP4 means within 4 nm along or across track. This would typically be used en route, whereas you would need RNP 1 or 2 around terminals, which are busy. RNP is self-monitoring and can warn you if you are likely to stray outside airspace boundaries, which are equal to twice the RNP value - RNP4 means an 8 nm wide corridor.