RNAV is an abbreviation for Area Navigation. There are two forms: Basic (or BRNAV) and Precision (PRNAV). The difference between the two is the degree of accuracy required (Precision being higher).
RNAV is a move away from the historic way of flying by reference to radio aids where you would just be able to track directly between ground based aids such as VORs or NDBs. This is quite restrictive, as it funnels all the traffic into one area or requires large numbers of (expensive) navaids to provide multiple routes.
RNAV allows you to fly perhaps a more direct route without passing directly between navaids, but maybe referring to them. So a position might be based on a certain radial at a certain distance from a VOR/DME, and you could have basic equipment which could take you directly there. Many light aircraft have this (KNS 80 is an example) and it is a good bit of kit. Effectively you can electonically 'move' the VOR/DME from where it is to where you want it to be, and you then use the cockpit instruments to track to that.
Further along the line, in the airways system we rely heavily upon the capability of aircraft to route directly from reporting point to reporting point without going via navaids to increase capacity and provide better flow.
Modern RNAV equipment might be the KNS80 style I mentioned above, or it might be GPS or INS or IRS, or a combination of several of these. The important point is that, where available, the RNAV equipment must cross check its assumed position with ground based aids, so it might automatically or manually take a cross cut from several DME readings to get a triangulation fix to confirm its position.
If you are looking at an airways map, you can spot an RNAV fix by the fact it has five letters (e.g. MARGO, SHAPP, LOGAN, CLIPY, NAKID) and a ground based navaid will have three letters (LAM, OCK, TNT, WOD, WCO) or two if it is an airfield locator beacon (NN).
This is a very basic explanation and I'm sure others will add more flesh.
The main point to get across is that RNAV allows you to fly a route which may be based upon but is not restricted to ground based aids.