Like any big, cosmopolitan city, Rio [1] is a great place for shopping, with something for most tastes and budgets. The best things to buy in Rio (as opposed to elsewhere) are Brazilian beach fashions and surfwear, antiques, and CDs. Precious and semiprecious stones are a classic purchase. You can also find traditional arts and crafts from all over the country.
Links:
[1] http://www.moon.com/destinations/brazil/rio-de-janeiro/rio-de-janeiro-city
[2] http://www.moon.com/destinations/brazil/rio-de-janeiro/rio-de-janeiro-city/shopping/shopping-malls
[3] http://www.moon.com/destinations/brazil/rio-de-janeiro/rio-de-janeiro-city/shopping/carioca-fashion
[4] http://www.moon.com/destinations/brazil/rio-de-janeiro/rio-de-janeiro-city/shopping/jewelry
[5] http://www.moon.com/destinations/brazil/rio-de-janeiro/rio-de-janeiro-city/shopping/books-and-music
[6] http://www.moon.com/destinations/brazil/rio-de-janeiro/rio-de-janeiro-city/shopping/antiques
[7] http://www.moon.com/destinations/brazil/rio-de-janeiro/rio-de-janeiro-city/shopping/arts-and-crafts
[8] http://www.moon.com/destinations/brazil/rio-de-janeiro/rio-de-janeiro-city/shopping/markets