RNAV is an aviation abbreviation for 'aRea NAVigation.' It refers to a method of navigating using a variety of navigation inputs to produce synthetic course information.
Okay, custo, what the hell does that mean?
In the Old Days (and still today, when you're training or in an older airplane) the navigation systems are primarily able to tell you where you are relative to a particular point on the ground. Those points may contain VOR stations, or NDBs, or DME transponders. In any case, the systems able to handle those types of navigation facility generally are excellent at getting you directly to or from that station. A VOR, for example, is easiest to use when you are flying directly towards (TO) or away from (FROM) a VOR station - the needle on the instrument in the aircraft, when kept centered, will keep you on whichever of those two courses you have selected.
This is reflected in the fact that to this day, all aviation navigation charts contain what are called airways - in the U.S. these are known as Victor airways -designated routes which directly connect these known points. (To learn what an airway means to air traffic controllers, see archiewood's excellent writeups.) This meant that in the old days (or, again, today in older aircraft) the easiest way to navigate was to select a route which took you from one of these points to the next, and so on along the surveyed airways until your reached your destination. This, while easiest technically, is less efficient both in terms of the distance travelled by each aircraft and also in terms of airspace congestion. If all aircraft are confined to these airways, the risk of collision is much higher, and traffic separation becomes more difficult.
RNAV arose when computers became sophisticated enough to perform continuous mathematical gymnastics aboard aircraft complex enough to display, to the pilots and the autopilot, where the aircraft was at any given time. This led to RNAV, where an aircraft can be 'told' via its Flight Management System (or just autopilot) to go directly from one location to another, or to follow a set of waypoints unrelated to surveyed beacons. The computer will use VORs, NDBs, or (these days) GPS, most likely, to continuously update the aircraft's current position.
RNAV has two components. They are LNAV and VNAV, and they stand for 'Lateral NAVigation' (basically, left-and-right, maintaining a given course) and 'Vertical NAVigation', which allows the aircraft to accurately determine its altitude and fly three-dimensional course guidance. Low-precision VNAV can be done using an encoding barometric altimeter, but of course these are subject to constant correction (below Class A airspace, in the US) as the local air pressure changes. This means that during takeoff and approach, for larger aircraft, this isn't feasible. Some FMS/autopilots can provide VNAV via DME slant range calculations, or via radar altimeter and elevation charts; some can use GPS for a full three-dimensional fix and provide it in that manner.
Note that LNAV-only RNAV is still quite useful, and allows the aircraft to directly fly its own course, or a course selected by an Air Traffic Controller without having to be limited to the surveyed beacon points. The words all instrument rules pilots hope for - "Bugsmasher XXX, cleared direct destination" are the result of RNAV, which allow the pilot to make his own way from point A to point B.
In practice, there are a large number of instrument flight 'fixes' - artificial waypoints on the charts. Some of these correspond to VORs, DMEs, NDBs etc. but many don't. They have five-letter names, and are generally pronounceable so that they can be listed quickly in voice communications. This leads to examples of aviation humor, as particular instrument approaches or airway routes can contain ordered lists of these fixes which may produce phrases or sentences. For example, there is a famous sequence near Pease, NH (which used to be an Air Force Base and is now a civil airport) which reads, in order if you approach from the Initial Approach Fix and transit to the airport: ITAWT ITAWA PUDDYE TTATT. If you miss your approach, you will need to head for another fix, and then loop around to approach it again - it is named IDEED.
If you're not a Tweety and Sylvester fan, this may not mean much. Near Las Vegas are POKER and CHIPS. Approaching Brisbane, Australia you pass LEAKY BOATS SINNK. There is another sequence near Australia's west coast which reads WONSA JOLLY SWAGY CAMBS BUIYA BYLLA BONGS UNDER ACOOL EBARR TREES. Make of that what you will.