The "brown" - or it is more often said, the "bowls o' brown" - are a stew made by various pot-shops in the alleys of the area of King's Landing known as Flea Bottom. Brown is often cooked for very long periods of time, simmering for year after year inside huge tubs where it receives new ingredients while being constantly served to the poorest residents of the city. Typical brown has a film of grease over it, and shows pieces of its vegetal ingredients; the meats are harder to identify, and most consumers don't want to try.
Common brown ingredients include barley, carrot, onion, turnip and whatever pieces of meat its makers can get hold of and dare to include in it, including fish caught locally. The meats may perhaps vary according to each pot shop, but it is not at all uncommon for them to include pigeons, rats, cats and even dead bodies of dubious origin.
It is reasonable to assume that hardly any meat is ever rejected by the pot-shops unless it is quite patently unfit for cooking; at one point Bronn claims that a pot-shop he knows uses "all kinds of meat" when asked to dispose of a murder victim. It is possible, and probably usual, for consumers to pay for their servings with caught animals (and presumably other stew-worthy foods such as many vegetables) that will in turn become fresh ingredients for the brown.
Brown pot-shops are dangerous places, always being watched by people of questionable reputation and motivation.
Despite its dubious sanitary and ethical qualification, brown is an important source of nourishment for the less fortunate among King's Landing inhabitants.
While the bowls o' brown proper are something of a cultural peculiarity of King's Landing, similar meals are not uncommon in other places of Westeros, sustaining the considerable numbers of people who can't easily find more pleasant food.