The Bastakia quarter, or Bastakiya, is an area of Dubai on the Bur Dubai side of the Dubai Creek that dates from the 1890s, where residents originally earned money from pearl diving. Bastakiya is named after the Iranian town of Bastak - many of the original traders in Bastakia were from Iran.
In the 1990s, most residents in Bastakiya had their homes bought by the Dubai Goverment, and starting in the late 1990s the Historic Buildings Section of the Dubai Municipality began redeveloping the area with old historic buildings renovated (or rebuilt), along with a number of art galleries and cafes being opened up. There are also a couple of pleasant boutique hotels with small doorways for tall people to bump their heads on. The area is one of the few historic areas in Dubai, and is also an excellent place to walk around for those interested in art & culture.
Many of the buildings in Bastakiya are topped with wind towers - a cooling tower that pre-dates the arrival of airconditioning, and is surprisingly effective in the summer heat of Dubai (at least that's what everyone tells the tourists, it's still hot though). http://www.dubaifaqs.com/bastakiya.php