Generally, specialists convert the 2 food/turn you have to feed them into "something else." The "something else" depends on the kind of specialist you have, the buildings you have, the type of leader you have, and the civics you're in. In any case, you need food surpluses to run specialists, and hiring specialists will decrease your city growth and/or cut down on the number of citizens you have to work non-food producing tiles, and they also count against your city's health and happiness limits (the exception is that if you have the statue of liberty and/or are running mercantilsm, each of which gives you one free specialist per city that you don't have to feed, and don't against you on health and happiness penalties).
One of the good things about specialists is that they provide three great person points. The effect of these is increased if you are a philosophical leader, you are in pacifism, or you have national epic or parthenon running (national epic has to be in that city). GP points eventually give you a great person, which have all kinds of useful uses.
If you are in representation, every specialist gives you three beakers (or three extra beakers if they were scientists), which makes specialists more attractive if you have that civic. If you have the Sistine chapel, every specialist generates (extra) culture.
One important thing to remember about specialists is that they generally provide near "bottom line" benefits, unaffected by the city's commerce or the commerce slider. For example, if I have a ton of merchant specialists in a city, they generate their money regardless of whether the commerce slider is set to 100 percent gold or 0 percent gold, and regardless of whether there are commerce improvements like cottages in the city. Their benefits ARE increased by Wall Street and other buildings (banks, etc), but are NOT increased by things that affect commerce (commerce terrain improvements, the slider, bueraucracy). So, the trick with specialists is to use them to provide things your commerce slider is not providing, and to put them in the city that has buildings specialized for that benefit (put merchant specialists in your wall street city, hopefully with a shrine or two; put science specialists in your Oxford city).