We went to so many great places. We were lucky to have a resident expat with us a lot of the time, so we could go to places the locals would go. I especially recommend wandering around the Barri Gotic and El Born. We popped into several places for a beer or for some tapas. We were never disappointed.
There is a place called Taller de Tapas in El Born right off the V. Latania (they actually have two locations). This place is a little pricey and slightly touristy, but the food was great. Get the sauteed duck foie gras.
There are also a bunch of places in which you pick, literally. There are a multitude of platters of small food, mostly on pieces of bread, stuck with a toothpick. You get a drink and a plate, and it is a free-for-all. Save your toothpicks, because that is how you are charged. There is a great place in the same square as Taller de Tapas and several other places all around El Born.
And remember, this is Europe--they take their time eating and drinking. The waiter will not bring you your check unless you ask; the food will come, but not quickly by American standards; coffee is meant to be sipped in the cafe (it is hard to get a to-go coffee, though Starbucks are sprinkled around the city). So be patient and relax. Also, at lunch (usually from 2-4pm), it is well worth it to get the fixed menu. This is generally three courses, and we never paid more than 14 euros per person.