The Tarima Basin has a long history and has been populated by different groups at different times.
Before the Uyghurs became a ruling tribe, they were under Dzungar Buddhists, who oppressed them. They were vassals of the Chinese emperors. Dzungars aggravated Chinese and then the Chinese massacred them, they then made Uyghurs rulers over the Tarima Basin.
Later on, Chinese conquered that region and made them subjects, but they revolted many times. Uyghurs has an independent nation very recently.
Xinjiang means New Territory in Chinese. East Turkestan is the name used by Uyghur nationalists led by Rabia Kader.