I'm an American who's been living in eastern Hungary for 6 months, and I go to Budapest at least twice a month. Thought I'd compile a list of my recommended eateries in the low to moderate price range for any visitors researching future trips.
These are not necessarily "destination restaurants" (places like Borkonyha, New York Cafe, Onyx have already been discussed) but great places to stop for a bite in between sight seeing or for a relaxed dinner.
1) Fricska Gasztropub. Located not far from Terror Haza, this underground cellar restaurant has a stellar lunch special -- 3 courses for <$8. An absolute steal for this level of refinement.
2) Castro Bistro. Near Deak Ferenc ter, this place has kind of a funky, beatnik vibe, and really inventive dishes. When I was there, I had an elk goulash in a peppercorn and blueberry sauce ($11). Really delicious, although a bit one-note after a while. My boyfriend's dish was a burger ($8) with veal patties, sheep milk cheese, dill, red onions, salted caramel croutons in a demiglace sauce-- the complexity of flavors kept it intriguing to the last bite.
3) Mazel Tov. Near Blaha Lujza ter in the midst of a lot of the ruin pubs (down the street from Fogas and blocks away from Szimpla Kert), this place is worth going just for the space. Hearty Israeli/Mediterranean food (pita sandwiches, shawerma, shakshouka etc) with an elegant ambience. There's live music most nights.
4) Hummus Bar, multiple locations. This chain is expanding. Excellent hummus, as the name suggests.
5) Karavan. This is a food truck market (multiple vendors) located right next to Szimpla Kert. Szimpla has a popular farmers market on Sundays, but if it's too crowded to eat there, I'd recommend going next door to Karavan and getting a langos (fried dough with a variety of toppings, the traditional being sour cream and cheese).
6) The 2nd floor food vendors I've tried at the Grand Market Hall (Nagy Vasarcsarnok) are hit or miss, and overpriced (by Hungarian standards). Also, tends to be crazy crowded. Instead, I'd recommend crossing the street and going to Leves, a take-away soup stand or heading over to Belvarosi Disznotoro which is not to far off of Vaci Utca.
A note on tipping:
Check your bill to see if service charge is included. If not, it's customary to tip up to %10, but never leave it on the table. Instead, hand it directly to your server. Servers generally bring you the bill with a wallet or credit card machine in hand so that you can settle the bill on the spot.