My interests lie in Mongolia, and my knowledge of Tibetan history is vague at best, so here goes:

The origins lie in the Qing Dynasty, which fell apart in 1912. The Qing, whose rulers were Manchu from the north, had subjugated Inner Mongolia, Outer Mongolia (Mongolia), Tibet, and Zungaria (Xinjiang) during its almost 300 year history. However, during most of this time Outer Mongolia and Tibet were kept largely autonomous, with Han Chinese settlers forbidden. Therefore, the cultures remained intact, and when the Qing dynasty began collapse, Mongolia and Tibet declared independence, citing that the old agreements of vassalage to the Qing rulers were now null and void. Obviously, the new Republic of China considered itself to be the successor to the Qing, and considered these territories as their own.

With all this turmoil going on, and no international recognition of Mongolia's independence, some turned to the newly forming Soviet Union (RSFSR) for help. The Soviets installed a communist regime in Mongolia, and drove out the White Russian (imperialist) garrisons that had invaded Mongolia. Of course, Russia was still in the process of consolidating power, and still had negotiations with China that affirmed Mongolia as a autonomous area under the suzerainty of China. At the closing stages of WWII, during the Yalta Conference, Stalin held recognition of Mongolia's independence as a bargaining chip in return for the USSR's declaration of war against Japan. Hence, by the end of 1946, Mongolia had become internationally recognized as a sovereign stage (albeit in reality a satellite state of the USSR). When the communists defeated the Kuomintang during the Chinese Civil War, the Kuomintang (or the former Republic of China) withdrew its recognition of Mongolia, but the newly created People's Republic, friendly with the USSR at the time, re-recognized Mongolia's independence in 1949.

However, admission to the UN was still not obtained. The ROC, now in Taiwan, still held China's seat on the Security Council, and in fact used its veto just once, to block Mongolia from admission (in 1955), as it considered it Chinese territory. In 1961, the Soviet Union forced the issue against the ROC, holding the admission of Mauritania hostage against Mongolia's admission. With pressure from other countries (especially African), the ROC relented.