Whether you are looking for leather items, books, clothes, souvenirs, or shoes, this city is a shopper’s heaven. You have a lot of options, depending on what you want to buy and especially on how much you want to spend.
For really cheap clothes, go to Once neighborhood. The exchange rate makes Buenos Aires a really cheap place for shopping, but this neighborhood is particularly cheap. Of course, the quality of the clothes isn’t the best, but you can still find good stuff, and it’s a 10-minute taxi ride from the center of the city. Shops are generally open Monday through Friday, and some of them are open on Saturdays until 2pm.
Price-wise, a second option would be Santa Fe Street. There are a lot of shops there, prices are still OK and the quality of the clothes and other items is better. Most of the shops here are open Monday through Saturday until around 7pm.
If you are looking for shopping centers, you also have interesting options. I have only been to Galerias Pacifico, located in Florida Street, which is a good place to buy stuff, but I have heard that Alto Palermo Shopping Center is much better.
And last but not least, you have Florida Street, where lots of tourists go for leather items and souvenirs. Most of the shops here are open seven days a week, and you can also enjoy street performances, a real plus! We went there on our last day in town to spend our (at this time, really few) pesos left.
My advice: wear comfortable shoes! You will end up exhausted after a day of shopping.