In 1971 the People's Republic of China (mainland China) replaced Taiwan (also known as the Republic of China) in the United Nations.
The independent papal state of 771 people (including the Pope) was created in 1929. They have not chosen to become part of the international organization.Vatican City - Because of its tiny population and unique status, the Vatican City is a permanent observer state, but not a full member. ()
Palestine - Attempted to gain membership in 2011/2012, but failed due to lack of support from the Security Council. The US would have likely vetoed membership. It is currently recognized as a non-member observer state. ()
Taiwan - The UN considers the People's Republic of China and the Republic of China to be a single state, therefore Taiwan is not a member.
There are several partially recognized states like Kosovo, Abkhazia, South Ossetia, and the Sahrawi Arab Democratic Republic, which are not members and, as far as I can tell, have not yet attempted to gain membership.