Most of the visitors from far away end up shopping in Pest in the middle of the city: Váci utca and nearby. It is historically the most expensive part of the city. You'll find Hungarian linens and lace, pottery, and other items, in souvenir shops.
You definitely want to visit the Great Market Hall Nagy Vásárcsarnok at FÅvám tér the recently renovated markethall with essential atmosphere it's at the south end of Vaci. Prices for the same items vary a lot between sellers and aren't set in stone so be sure to compare and bargain.
non-speciality shopping
Also, chain stores can be found along the Váci utca C&A, H&M, Clinique, Estee Lauder, New Yorker, etc.
The shopping malls locally known as "Plazas" are usually good for buying clothes, but prices may vary wildly even in shops next to each other. For electronics, the cheap supermarkets like Electro World and Media Markt are good targets, but the quality is on par with the prices. Due to the low cost of labour, a tradition in repairing mobile phones and other appliances exists, and buying second hand electronics is normal. This service is usually offered in smaller private shops.
Absinthe is available for purchase at common liquor stores, a must-have purchase for the European traveler. Many brands available in the Market Hall and liquor stores are of poor quality or not even "real" Absinthe.