Here are some of my notes from our trip a few years ago, so don't know the current status of these places. Some of my comments have references to NYC (where I live).

Petra - Carrer de Sombrerers 13 (Borne, next to St. Maria del Mar -church). Found this by just wandering. Menus were labels on wine bottles. Chandelier's had silverware hanging from them. Very (Greenwich) village-y kind of place and good food.

L'Oliva - Carrer Jovellanos 2 (around the corner from Hotel Inglaterra, where we stayed). Great 12E lunch menu, lots to choose from. Also supposedly good for dinner. The street is easy to ignore (not particularly interesting), but we just wandered down there because it was so near the hotel.

Cerveceria Catalana - Carrer de Mallorca 236. Great tapas, hopping spot. The indoor wait on a friday evening was long (1.5 hrs), but outdoors it was only 1/2 hr. It was worth the wait, and while waiting you can order a drink and tapas, at the bar to get started. This is in the Eixampe area.

El Gran Cafe - Carrer de Avinyó, 9. We had a very nice dinner here, loved the Escalivada appetizer. This is the more typical looking place you see all over, but this one was particularly good.

7 Portes - big seafood place down by the beach. Looks a bit like balthazar in NYC (brasserie-ish). It was good but not great. It's written up in many of the guide books.

Les Quartre Gats - don't eat here, but poke your head inside, or maybe just have a drink. Food is lousy but has history of Picasso hangout, etc. If you go on a Picasso tour, they'll definitely bring you here.

Senyor Parellada, adjacent to Banys Oriental hotel in Born area, C/ Argenteria, 35. This is more upscale than the others. Very civilized. More 'grown-up' than the funkier joints.

For more info on restaurants and just about everything else, print out Maribel's guide at http://www.maribelsguides.com/

I just want to add the 7 Portes and Les Quartre Gats are both very touristy.

Set Portes and Els Quatre Gats may get their share of tourists.
But at least 7Portes also has a fair share of locals. Quite famous for their paellas.
4gats should be more seen as an "event" , not just a place to eat. A bit like the Deux Magots in Paris. So definetely touristy. Just as PP described.
Just IMO.
Another good, but pricey place for sea food would be the restaurant Barceloneta.

Mixed opinions about 7 Portes, but on the informative 11870.com-site Spanish customers seems to be very satisfied, generally describing it as "un clásico". http://11870.com/pro/restaurant-7-portes

Lots of history in the walls. The greatest flamenco dancer of all times, Carmen Amaya from the close by Somorrostro district (Barceloneta), used to perform here as a young girl together with her father, and Federico García Lorca - possibly the most important Spanish poet and dramatist of the twentieth century - resited his masterpiece Romancero Gitano (Gypsy Ballads) for the first time in public here.

Carmen Amaya: http://www.youtube.com/watch?v=FCca6d2ceI0&feature=related
Lorca and Gypsy Ballads: http://www.spainisculture.com/en/obras_culturales/romancero_gitano.html

I also recommend Cerveceria Catalana, as well as Ciudad Condal in Barcelona.