The Random Number Generator, or RNG, is a mechanic that, obviously, randomly generates numbers. This mechanic is almost key to the Fire Emblem formula, producing two numbers when one engages in battle: hit and critical. The hit percentage is out of one hundred. If the hit number was 75, and the RNG produced 74, then it is a hit. If it produces 76 or higher, then it is a miss. The same process is used for determining criticals, but a solid hit is usually required in order for one to be possible. RNG is also used to describe what stats a unit gains when leveling up or describe forms of "infinite items" tricks in Fire Emblem: The Sacred Stones.
Note that in all games released outside Japan starting from Fire Emblem: Rekka no Ken, the Random Number Generator generates two numbers instead of one for hit chance, and the average of the number is the true hit value; in other words, a displayed hit (hit shown on screen) of 75 will actually hit on any two numbers whose average is less than or equal to 75. The effect on overall accuracy of attacks makes a "bell curve" effect: it becomes exceedingly rare for an accurate attack (90+ hit) to miss, while inaccurate attacks (Hit lesser than 50) will land far less often than its hit rate would suggest. This is beneficial to the player, as their units usually have higher hit rates while the enemy units tend to have lower hit rates. However, for Fire Emblem releases that are released before Rekka no Ken, like Fire Emblem: Binding Blade and translated into English by fans of the series, the RNG hasn't been changed and as a result, enemy units may seem more likely to hit the player's units.