Mostly the markets are open in every Saturday between 6:30AM-2PM. However the biggest markets like Lehel square, Fehérvári streets are open from Tuesday until Saturday during the day.
Please be careful and avoid the Fővám square market. It's a tourist trap unfortunately.
You can buy fresh products here like
eggs
breads and cookies
Hungarian paprika powder, chili powder, green herbs
local honey and fruit syrup
sour cabbage - must try it
meats, sausage, töpörtyű
different types of the nuts and beans
of course seasonable vegetables and fruits and flowers