After the takeover of mainland China, the US supported the ousted Kuomintang, and it was the Republic of China (based in Taipei) that represented China in the UN (due to the influence of the US and the West).
The USSR supported the People's Republic of China. They boycotted the UN during 1950 in protest at Mao's exclusion, which allowed the US to gain the backing of the UN in the Korean War (the USSR could have vetoed this had it been in the session).
Brinksmanship and sabre rattling continued from the mainland across the strait. In response, the US pledged military support for Taiwan should she be invaded. Following the Sino-Chinese split in the late 1950s, the US began developing diplomatic relations with the People's Republic. But it was not until 1976 that the PRC replaced Taiwan as the official representative of China in the UN.