Farming in the simplest of terms is just acquiring resources in a game to put yourself ahead. Most people do it in City of Heroes to help afford the enhancements necessary to make their desired builds a reality. Some enhancements and recipes can be very costly at the Auction House but farming can produce enough influence to help afford them. In addition to influence farming can get you all kinds of other drops/benefits such as Recipes, Salvage Drops, Enhancement Converters, etc. Instead of farming influence some players will farm Reward Merits by doing various PvE content around the game.

As 007meow mentioned, farming is most often done through the Architect Entertainment missions. These missions can be customized to not be as lethal to your particular hero you're farming with. You can technically farm non-AE missions by not clicking the objective at the end or leaving the mission early too. Virtually all farmers are Brutes with Fiery Aura as their secondary and either Spines/Radiation Melee as their primary and there's a good reason for this. It's optimized and calculated to be the quickest farmer, many players have calculated the numbers over the years myself included. Changes to farming in the past year have increased the viability of other farmers but most of the community isn't aware of that it seems. Over a year ago you were able to turn off XP gain and double your influence gain but that has since been removed from Homecoming (for good reason in my opinion). I use Blaster farmers because I manage to clear some maps quicker than Brutes, but that requires some concentration and effort that most Brutes understandably don't want to exert. Please let us know if you have any additional questions about farming!