Bucharest is home to a number of markets, most of which stock a wide range of produce, at prices often (but not always) cheaper than the supermarkets.
In season fruit and vegetables tend to be local, although it depends on the market: check what you are buying if you really want to make sure you are getting local produce. In winter, expect almost everything to be imported and priced accordingly.
The biggest of Bucharest's market is Obor: an amazing place which is essential for anyone whose quintessential idea of Romania is women on street corners selling turnips. Read about the place (and its dark past) in full here.
Bucharest In Your Pocket - for what it's worth - tends to do its shopping at either Piata Ramnicu Sarat or the Peasants' Market on B-dul Unirii.
Beyond the produce markets there are a couple of bazaar-type places worth visiting, not least Complex Comercial Rahova, a place that stocks cheap everything. Take tram No. 32 from Piata Unirii to (almost) the end of the line to find it. A similar den of mayhem and chaos is the Sunday morning flea market at Vitan-Barzesti.
The big Flower Market on Calea Rahovei is also well worth a visit.
Also look out for arts, crafts, peasant and antiques markets too: these are quite common at the weekend. The Peasant Museum holds one roughly every two weeks, while there is a small antique market every Sunday at Parcul Carol.