RNG stands for "Random Number Generator". In the context it's used here, it means the process by which Pokemon are either bred or captured with ideal IVs, shinines, gender, ability, ETC by abusing the psuedo-number generator used by the Pokemon games.
So when someone talks about "RNGing" they're talking about obtaining perfect Pokemon in game by abusing the way the RNG in Pokemon works.
Basically, stuff that happens in the Pokemon games are never truly random because the number generator the game uses to determine random events doesn't generate completely random numbers; instead it, produces them using a very complex formula and process that starts with a number that is the "seed" for a game. This seed is set by the date and time of the DS (or in Emerald, the time set on the in game clock) when the game is is started. Using the fact that these seeds can be determined by recording the time at which your game is turned on, a program called RNG reporter has been developed that lets you figure out times at which to start your game that will cause you to encounter so called "perfect" Pokemon.
While much of the process is simplified by RNG reporter it's still a pretty complicated process. I won't get into it fully here, but if you're really interested this set of articles explains more on the process. If you're using a simulator knowledge of this process is pretty much completely irrelevant, but if you're looking to battle in Wi-Fi RNGing your Pokemon can give you a very important edge over your opponents.
