Farming is the practice of repeatedly performing an action in order to receive drops -- gold, reagents, equipment, etc. Players will farm areas for reagents (sections of Regents Square tend to have scrap iron and mistwood, for example); bosses for specific pieces of equipment (e.g., Malistaire's Dragonfire Cloak); or bosses and other NPCs for sell-able equipment.
Farming reagents involves finding a relatively small area in which they are likely to appear (or "spawn" in game terminology), running through that area and picking up as many as appear, then changing realms and repeating the process.
Realm switching simply means pulling up the "Realms" section of the game's main menu and picking a realm from the list, usually one listed as "Perfect" or "Normal" to reduce the chances of running into another farmer. There is a 60-second wait required between realm switches, so the ideal farming route lasts about that long.
One of the better examples of a farming route is that of scrap iron, which can often be found on the sidewalks of Regent's Square between Digimore Station and the Royal Museum, which takes about a minute to cover while on a typical mount.
Bosses often drop high-end equipment which can either be worn by the player or sold to a vendor (or in the Bazaar). Thus they are popular targets for farming.
Often-farmed bosses include Kraken on Triton Avenue in Wizard City, Jade Oni in MooShu, and Kraysys in Dragonspyre because they can be accessed directly, i.e., without having to make your way through a dungeon first.