"Belly Dancing" originates from the folkloric dances of the Middle East. Some of these dances were religious or spiritual in nature, others designed for court or harem entertainment, others celebratory.

Regional and cultural variations have given birth to many different dance styles, each with distinctive music and costumes.

The style of belly dancing most Westerners recognise is a relatively new (20th century) fusion of many different Middle Eastern dance steps with a generous dollop of Hollywood and western ballet steps thrown in!

Other Names for Middle Eastern Dance:
Belly Dance, Bellydance, Oriental Dance, Raqs Sharqi,  Danse Orientale, Tsiftitelli, Egyptian Dance

Countries of Origin:
Egypt, Lebanon, Turkey, Iran, Iraq, Algeria, Tunisia, Morocco, Saudi Arabia, Greece, India etc.  

Other people have already written extensively on the origins of Middle Eastern dance so here are some links:

Origins of Oriental Dance
In Search of the Origins of Dance
Why the Fuss over Egyptian Style Music and Dance?