Officially: No, Mongolia is not a part of China.
Mongolia has their own language, prime minister, parliament, and president.
What is known as 'Inner Mongolia' on the border of Russia and the independent state of Mongolia is an autonomous region belonging to the People's Republic of China.
Mongolia declared their independence in 1911, however, the Republic of China had other plans for the region. Chinese forces occupied part of Mongolia until Russia invaded in 1920. Russia decided to support the creation of an independent, communist government in Mongolia. With the Soviet Union's help, Mongolia once again declared their independence -- ten years after the first attempt -- on July 11, 1921.
Only in 2002 did China stop considering Mongolia as part of their 'mainland area' and removed it from maps of their territory.
Despite the struggle for independence and world recognition, Mongolia still somehow maintains good international relations today with the U.S., Russia, China, Japan, and India -- countries that often have conflicting interests.
Today, Russian is still the most widely spoken foreign language in Mongolia, but English is spreading widely.