Burano has been famous for making high-quality lace since the 16th century. Although lace isn't as popular as it once was, lace making has been revived as an artisan craft and you can sometimes still see lace being made by hand. Lace products are sold in many of the shops but check to see that it's handmade and not mass-produced for tourists. If you want to buy artisan lace, expect it to be more expensive.
A good way to find out about Burano's tradition of making lace is to visit the Lace Museum, Il Museo del Merletto di Burano, housed in the former lace-making school dating from 1872. Exhibits include the origins and history of lace making starting in the 16th century, samples of lace, and the Burano Lace School. In addition to lace, there are paintings and drawings and displays of period dress and glass. There's also a video and usually there are artisans demonstrating lace making.
Lace Museum Visiting Information - Museo del Merletto di Burano
Location: Piazza Galuppi
Hours: Opens daily at 10:00, closes at 18:00 April - October and 17:00 November - March. Closed January 1, May 1, and December 25
Admission: 5 euro (as of 2012). The Lace Museum can is also part of the 11 Museum Venice Pass which can be bought at museums or purchased ahead of time in US dollars from Select Italy (see information or buy a pass).
Website: Museo del Merletto (in Italian)