Some of the establishments recommended here are covered in major guide books while in the others most customers are locals rather than tourists. They serve good food at budget to mid-range prices. Some high-end restaurants are included too.
General tips
Hungarians love meat and cream. Many types of sauces actually contain loads of fresh cream (fözötejszín) or even lard, which makes them rather heavy. Keep that in mind when ordering food to avoid unnecessary stomach-ache ;-)
Some useful words
alma = apple
bor = wine
burgonya = potato
csirke = chicken
csirkecomb = leg of chicken
cukor = sugar
eper = strawberry
étterem = restaurant
fehér = white
jugtúró = cottage cheese made of sheep milk
kenyér = bread
köröm = trotters
lecsó = a sauce made of paprika, onions, smoked meat...
marhahús = beef
meggy = sour cherry
méz = honey
pulyka = turkey
sajt = cheese
sertés = pork
sör = beer
szolo = grapes
tehéntúró = cottage cheese made of cow milk
tejföl = sour cream
tojás = egg
víz = water
vörös = red (used for wine)