Basically, a nation RP is a roleplay where you're controlling an entire nation/organization rather than a single character (hence the name), although this may be embodied in the ruler of said nation/organization as your "main" character. I think the easiest way to put it is... Hmmm... Basically, if other RPs are like D&D, NRPs are more like Civilization, I guess? If that makes sense.

As to how this is pulled off, it depends on the GM, really. Some like to make big expansive gaming systems with lots of number crunching and point builds and stuff like that to enforce a semblence of fairness to the whole thing, whilst other prefer a more narrative and less mechanical approach. Both have their merits, really, so it just really depends on the GM and players which one they go with. Hope that helps.