RNG is a game term that is an acronym for "random number generator". Random number generators (or, more correctly, pseudorandom number generators) are algorithms used in computing when an outcome needs to be as random as possible. A computer can't generate completely random numbers on its own, so it uses a complex algorithm designed to get as close to random as is practical. These algorithms are used in the game any time an event has a probability of happening - whether that be critical hits, the spread of weapon damage, procs such as Missile Barrage or Sudden Doom, or to determine which loot drops from a raid boss.
The term "RNG" is frequently used in conversation to mean "luck" or "lucky", as in "bad RNG" and "good RNG". For example:
In this post, Ghostcrawler is contrasting the damage reduction provided by armor and by dodge and parry, which are collectively called "avoidance". Armor is a set reduction to all incoming attacks - a 60% armor value means that all melee attacks are reduced by 60%. Dodge and parry, on the other hand, are a percentage chance to completely avoid the attack and take no damage from it. It is possible to be unlucky (or get "bad RNG") and have a streak of hits that are not avoided, which can cause very high incoming damage and may cause sudden death.
RNG is frequently viewed in a negative light because of this "bad RNG" effect, although it can be equally possible to get "good RNG" that results in a favourable outcome such as a long streak of blocked or parried attacks. In a raid setting where bad luck can lead to a wipe, some raiders prefer the certainty of non-random abilities to the possibility, however remote, that bad luck might cause disaster.
"Bad RNG" and its unpopularity with some players was acknowledged by Jeffrey Kaplan at the 2009 Game Developer's Conference, where he talked about "progressive percentages" in quest item drop rates.
We found that this had a lot of problems where players would run into streaks, and they only remembered the shitty streaks. So what we decided to do was we took a page out of Warcraft 3, which had a very elegant design which they referred to as 'progressive percentages.' ...In Lich King, every creature that is part of the collection quest has the item 100% of the time, but we do a progressive system where we up the chance the player [gets the item] each time he kills it. The problem was, when we put it live in the beta--and we didn't tell anybody this--we found that while it was great that it got rid of the bad streaks, it also got rid of all the good streaks. And overall we needed to raise the base drop percentage to around 45%.
The elimination of the "bad RNG" factor of killing hundreds of mobs for a single quest is important, especially given the difficulties of preserving the effects of "good RNG" at the same time. This topic is discussed in more detail in the paper Linear Regression of Progressive Percentage in World of Warcraft.