The key to good seafood is freshness, and Boston gets nothing if not fresh fish. Almost any bar will do great fish and chips for 10 bucks or less. Ditto fried clams (the real kind, with bellies). Woodman's in Essex claims to have invented them.
The Barking Crab is fun in summer, on the younger and louder side. Food is good though. They were closed for a spell a few years back when they were caught filling their lobster tanks with water right out of the harbor. A few patrons got sick as a result and many locals now refer to the restaurant as "The Barfing Crab." They're located right next to the Moakley courthouse on the waterfront.
Not far away is The Chart House at Lewis Wharf, which is a chain place located in a gorgeous 300+ yo building. The food is uninspired but acceptable, and you can take the Logan Water Taxi to Rowes Wharf which is a short walk away.
In Faneuil Hall, the Naked Fish and Kingfish Hall get good reviews. Also worth a try is the (in)famous Durgin Park which, despite being a tourist trap, still draws a local crowd due to large servings of decent slop at low prices.
Finally, while not a "seafood place" as such, I have to plug the legendary Locke-Ober in Downtown near the Common. It has been around since 1870-something and shows little evidence of change. The menu has a bit of everything from steaks to seafood but they are especially well know for two lobster dishes- baked Lobster Savannah and JFK's Lobster Stew. For $18 you get a hefty bowl of bisque-like broth with a few ounces of fresh lobster and about a gallon of sherry. It is almost a meal on its own, and well worth the price. The dining room is amazing and it makes you feel as though you've been transported back to the Belle Epoque. Be aware that jackets are required, but last time I checked they'd loan you one if necessary (which to me is the classiest move around).
Now that's all I know about seafood, but if what you really want is a good watering hole, let me know, I'm an expert on that...
-cwk.