There are several streets in Budapest which are packed with shops. From shoes over clothes to electronics and so on you can find anything in the streets of Budapest. I think there is no real city center in Budapest so that over time it came that in some streets a lot of shops opened and by now they are streets to shop. The most famous are listed below.
Andrassi ut is the most famous street in Budapest. It starts at Deak Ferenc ter and heads towards Hösök ter(hero square). On its way it passes Budapest highlights like the Basilika, the Opera, Liszt ferenc ter andOktogon. On Andrassy ut many shops, interior design shops, restaurants and bars are located. Famous is also the Christmas decoration on Adrassy ut.
Vaci utca starts at Vörösmarti tér and leads to the market hall. On Vörösmarti ter is the world famous coffee house Gerbeaud.
The street is a famous place for tourists with all the shops, restaurants and cafes. At nighttime many tourist like to take a walk there but beware because many “nightclubs” offering striptease shows try to get you in the club which ends always in a disaster as people say in the internet who have visited those locations. The nightlife clubs send out their ladies (in groups of 2) to guide the tourist in a bar.
But do not hesitate to have a walk there, it is really nice.