It's always best for people to define themselves, and so I have taken the italicized information directly from the Uyghur American Association website.  Note again there are several spellings of Uyghur.

Uighurs, or Uygurs, are a Turkic-speaking people of Asia who live mainly in northwestern China in an area named Xinjiang province by the Chinese government. 

Xinjiang is a large, sparsely populated area that has been a site of heavy army and police concentrations since 1949, and is used as a base for nuclear testing, military training, and prison labor facilities. The population of 18 million includes several Turkic-speaking Muslim ethnic groups, of which the Uyghurs, numbering eight million, are the largest. The percentage of ethnic Han Chinese in Xinjiang has grown as a result of government policies from six percent in 1949 to 40 percent at present, and now numbers some 7.5 million people. Much like Tibetans, Uyghurs in Xinjiang have struggled for cultural survival in the face of a government-supported influx by Chinese migrants, as well as harsh repression of political dissent and any expression, however lawful or peaceful, of their distinct identity.

The Uyghur people call this area, their homeland, East Turkestan.  Use this name in China and you will get into a lot of trouble. 

After Chinese Nationalists overthrew the Manchu Empire in 1911, East Turkistan fell under the rule of the Nationalist Chinese government. The Uyghurs, who wanted to free themselves from foreign domination, staged numerous uprisings against Nationalist Chinese rule and twice (once in 1933 and again in 1944) succeeded in setting up an independent East Turkistan Republic.

In 1949 the People's Republic of China was founded and East Turkestan came entirely under Chinese rule.  The republic was banished and the name of the area changed to Xinjiang Uyghur Autonomous Region, also known as Xinjiang province.  The direct translation of Xinjiang is New Territory.