If you would like to sample some traditional Slovak fare or some of the Pressburg specialties, though they are not what most people eat anymore, try Prešburg on Michalska. Another option is the Traja musketieri styled as "the taste of 1625" on Sladkovicova (not far from the Presidential Palace). The taste of traditional Pressburg, which draws on Austrian and Hungarian culinary traditions is offered in Leberfinger, which is across the Danube from the Old Town (a pleasant 10-minute walk across the bridge and most probably a taxi ride back as you may be too full to walk).
Another restaurant with traditional Old Bratislava fare is Modra Hviezda (The Blue Star). It is located on the narrow Beblaveho street, which is a road that leads to the castle and serves meals in the Slovak, Austrian and Hungarian traditions. The food is accompanied by a great selection of Slovak wines and a bit of historical input is written on the menu.
Staroslovenska krcma (the Old Slovak Pub) is in the pedestrian zone on Michalska and is another solid choice, both in the winter when you eat in their tastefully decorated cellar and in the summer at one of their two outdoor seating areas.
To eat where the locals eat, visit Lahodky u Sherlocka on the corner of Mileticova and Prievozska. Another option is one of the slightly more upmarket, but still very reasonably priced restaurants, such as Steam & Coffee. Steam & Coffee is a local chain which can now be found at near the Polus City Center and at the Slovak Radio upside down pyramid building.
If you are interested in the more exotic restaurant offerings (Chinese, Indian, Japanese, Middle Eastern...), see our Ethnic Restaurants in Bratislava listings.