What is the difference between RNAV and RNP? I know that RNAV doesn't need monitoring and alerting, but RNP does.
RNP is a superset of RNAV requirements. It applies and communicates a performance specification for RNAV that may vary by location rather than by equipment.
On board performance monitoring in RNP requires an alert when the system cannot guarantee with sufficient integrity that position meets required accuracy. Required accuracy can vary. A tight approach may require 0.1nm accuracy while 5-10nm over oceans might be acceptable. RNP requires communication of a numeric qualifier for accuracy. RNAV has no such mechanism.
Confusion is understandable as the terms are sometimes used interchangeably. ICAO is arguing with FAA and other countries to rename RNAV approaches as RNP approaches. The ICAO position is more clear, the FAA position supports existing equipment and training. Everyone believes a navigation performance spec needs to be communicated to pilots in charts and equipment, but there is no clear answer as to how to do it at this point.
RNAV, or Area Navigation is a means of navigating that allows direct routing between any two points on the earth. This is opposed to VOR only navigation that requires following routes that pass along radials of the VORs. An FMS or GPS navigator are both RNAV capable systems.
An RNAV procedure is one that requires you to have an RNAV system in order to fly it.
RNP, or Required Navigation Performance adds specific performance standards onto RNAV systems. It defines several levels of performance that are applied to specific airspace and operations. The need for these levels is driven by airspace management and air traffic control (keeping planes from running into stuff or each other).
The most visible part of RNP is the monitoring and alerting requirements. An RNAV system can compute your aircraft position to some level of accuracy. With RNP the system must estimate its actual navigation performance (ANP) and compare it to the specified RNP for the airspace or operation you are flying. An alert is generated if the RNP is not being met.