A profession is a craft or tradeskill that player characters may learn regardless of their character class, faction, or race. The first recipes you get are useful for gearing up low level characters (assuming a higher level character is not helping to support you). Some contend that as soon as you start running instances, the drops will usually be better than most crafted items from the same level, but this is not always the case. Quite often crafted items will provide comparable stats or utility benefits that are quite useful for characters of all levels.
The most benefit to your character comes from trying to keep your professions leveled up as you progress through the game. High end crafting, including specializations, can be extremely useful and lucrative, especially from patterns that come from end game faction grinding or drop in high level instances (some of which Bind on Pickup). There are also several quests which require crafted items for completion.