I've lived in Bratislava for about a year. Here's a list of some of my favorite go-to's, recently compiled for an acquaintenance and further refined.
A caveat that this is not an all-inclusive list. These are just a few places that I like to patronize and which are reliable and easy on the wallet in comparison to what you generally find in Old Town.
Traja Musketieri - on Sladkovicova ulica. (not far from the Presidential Palace) has some of the consistently best Slovak style food, including my favorite garlic soup (broth based). Very hearty food and good service in a charming but relaxed environment. Reservations almost always required.
U Mamicky - also close to the Pres. Palace on Palisades ul. The chef/owner is a transplanted Croat and his specialty is grill cooking - fresh seafood, beef. If the weather is still warm he has a wonderful garden in the back. His small dining room is overly decorated (peach-colored table cloths and napkins and kind of frilly) IMO but there's no dress code. Excellent place. He'll cook whatever you'd like but he likes to make recommendations based on what's fresh. He often makes the drive to Croatia to bring back seafood that you can't otherwise find in SK.
Leberfinger - on the other side of the bridge from Old Town but a nice walk across the River Danube. Traditional slovak food and a relaxed, comfortable environment.
Slang Pub - across from the Radisson on Hviezdoslavovo namestie. Good beer and an extensive menu. Definitely a dive but if it's a weekend, reservations are a very good idea. Their bbq pork ribs are really good though much different from what you'd find in the US. Nice covered summer patio.
Slovak Pub - up a flight of stairs on Obchodna ul. A dive but offers an extensive menu of traditional Slovak food and beer at very good prices. The Euro Pub right below it has pretty good pizza though there is no shortage of pizza restaurants in Bratislava, many of which cook in brick ovens.
Ludwig's - Old Town. Really good value and food at a white table cloth restaurant in a beautiful building. If you want to dine at a high end place I would opt for Ludwig's over the more famous Le Monde, which is part of the Czech-based U Kampa restaurant group. IMO Le Monde is inconsistent and their service can disappoint as well.
Other recommendations -
Paparazzi - up-scale italian in Old Town. Great menu, expensive wines. Nice cocktail bar. Always draws a crowd of tourists and locals.
Al Dente - italian in Tatra Centrum next door to the Crowne Plaza. Very quiet place in a shopping center environment, but good food and easy on the wallet.
Steam and Coffee - a local chain. I avoid the one in Old Town next to the Radisson because I find the other locations have a better environment, food and service. But the location at the Slovak National Radio (SNR) building is very good (walkable from Old Town, or about a 5 min walk from the Crown Plaza), as is the location on Mileticova near the outdoor market.
Chez David - a mix of jewish and slovak style food (not kosher) behind the castle on Zamocka ul.
Samir - lebanese. Perched on a narrow road as you walk up the hill toward the castle.
Kaffe Mayer - coffee and pastry in Old Town. Best place in town for sweets and cakes, especially the streudel and puncovy donut.
Cokolada - a great place to stop in for a hot chocolate. They have at least 30 different styles. Old Town.
My most cravable food in Bratislava is losos and the best I've found are at the outdoor market on Miletičova ulica. The market is known as centralne trhovisko and is a fascinating mish-mash of a farmer's market, flea market and floral stands. Losos are a fried dough concoction and I like to order it with garlic (cesnakom) or cheese (syr), or both. Look for the food stands just on the outskirts of the fresh produce area. You'll know you've found the right place because there will probably be a line. It's also right next to a hamburger stand.