Modern Tallinn has fine shops with prices to match but a taste of older Tallinn still remains at the central market, or Keskturg. See it soon: it is so close to the centre that it will surely be redeveloped as soon as the recession ends.
Cross over the main road and walk down a side street named Torupilli ots (literally Bagpipes Alley). You'll know you're going in the right direction if you see elderly people carrying sacks of produce towards you.
Look out for socks made in Belarus from flax. They're cheap, incredibly light and far more comfortable than cotton, the ideal bed socks.
There is a separate hall for meat, dairy products and other items that need to be kept cool. Smoked cheese or venison salami will make delicious souvenirs.
Drink it with soda (1 part juice to 4 parts water) for an incredible pick-me-up.