Inner Mongolia is a Chinese province whereas Outer Mongolia is an independent country called simply Mongolia.
Both Inner Mongolia were once provinces of the Chinese Qing Empire. In the chaos of post-Qing China and the Communist Revolution in Russia, the Bolsheviks came to occupy Outer Mongolia and set up a friendly, communist Mongolian government in the 1920's.
Russian power in the post-WW2 period was enough pressure to get both Republican and Communist China to relinquish claims to the province.