First, a disclaimer: my understanding of this stuff is cursory only. So this is just a basic overview. Check the TSO's for the latest and greatest.
Stands for "Required Navigation Performance". Essentially, it means that the aircraft's navigation systems are precise enough to keep it within a particular distance of where the system "thinks" it is 95% of the time. So for example if the system is RNP .3 that means the aircraft ACTUALLY IS within .3 nm of the computed position 95% (or 98%, 99% or whatever) of the time (or with 95%, 98%, etc. statistical certainty).
So there are these new GPS-type approaches where the autopilot flies the aircraft on a curvilinear path (a curved path to the runway), and to be allowed to fly them your aircraft needs to meet a particular set of avionics requirements (like .1 RNP or something, not sure). But basically the autopilot flies this curved path to the runway around mountains or through a canyon, etc.