From Diablo Wiki
Farming is a term meaning to repeatedly and/or easily clear a level in search of profit, as opposed to merely surviving. Players refer to "farming" or "farm runs" when speaking of fast, efficient killing speed, ideally with high MF and +exp for maximum profit.
Farming is a considerably different style of play than merely surviving, or taking on higher levels of Monster Power. Most farming builds are focused around fast movement speed and resource generation, and farming builds are generally built to do just enough damage to quickly clear on MP0 or MP1. Skills with higher resource cost are often used for their burst damage, when such attacks would not be very suitable against monsters that can not be killed off so quickly.
A common example is the Demon Hunter skill Multishot, which deals huge damage very quickly to a large number of enemies, but comes with a heavy resource cost. It's ideal for farming, when monsters can be wiped out in just a few shots, but less suitable for tougher enemies.
- Note that all variables exist on a sliding scale, since better equipment yields higher damage. Thus a very well equipped character can "farm" enemies that a character with worse equipment would have to grind against.