According to Chapter II of the UN Charter, in order to be admitted to the UN, a country must first be recommended by the UN Security Council and then approved by a vote of the UN General Assembly.
Admission to membership in the UN is regarded as an important indicator of sovereignty and legitimate statehood, especially for microstates.
Some states have governments that are unofficially recognized as independent, such as the Republic of China on Taiwan, but have not been admitted to the UN due to a veto by a P5 member.
When they began letting those countries join, it led to a period of great expansion of UN membership, especially as the number of colonies gaining independence increased.
Chapter II also provides for the suspension and expulsion of member states, and for the restoration of rights, by the UN General Assembly upon recommendation from the UN Security Council.