The Republic of China (hereafter "ROC" or "Taiwan" for brevity, but in modern terms Taiwan is just as appropriate) was in the UN from 1945 until 1971, as per Article 3:

The original Members of the United Nations shall be the states which, having participated in the United Nations Conference on International Organization at San Francisco, or having previously signed the Declaration by United Nations of 1 January 1942, sign the present Charter and ratify it in accordance with Article 110.

The ROC was there, they signed, they were members. This was revoked in 1971 with Resolution 2758 which "restored" the People's Republic of China's (hereafter PRC) right's as the representative of "China" in the UN. So now the historical background:

After the Civil War, ROC and the PRC had no diplomatic relations. They also refused to have any diplomatic ties with any country that recognised the other as "China". They effectively forced other countries to officially only recognise one of the other. Each claimed to be the only rightful government of Greater China. In part this was drawn out as the ROC, after fleeing to the island of Taiwan, fully expected their situation to be temporary, and believed they would retake the mainland in a matter of years. That turned into decades.

In 1971 just under 20 member states drafted a resolution which became Resolution 2758 (which you can read here). The United Stated made an attempt to preserve Taiwan's membership in the UN, if not as a permanent member on the Security Council, but this vote failed. the ROC was kicked out and now, despite being a separate nation, they are typically not recognised by UN member states, the ISO (International Organization for Standardization) and so on.

They do have unofficial relations, and there are "Taiwan Economic Cooperation Offices" in other countries in place of embassies and consulates. There's an ambassador role as well, though not under that name. They are for all intents and purpoes a separate country, in the same way that North Korea and South Korea are, except that many countries, while still doing business and selling weapons and having quiet relations, do not officially recognise Taiwan for fear of upsetting China.

Sidenote: In 2007 the ROC reapplied for membership under the name Taiwan, but were denied since "they're part of China", despite not actually being a part of China.