Countries are not part of UN

Following international law, all nations in the world are part of the United Nations, therefore there would be no nations that are not part of the United Nations.

However there are five "difficult" cases in this. Namely Palestine and Kosovo, which both do not have enough international support to become a United Nations member, and for Palestine it is a controversial topic by itself whether it is a separate country or not. Taiwan left the United Nations in 1971 when the Republic of China got a seat. South Sudan recently became independent and therefore has not yet applied to be a member. And the Vatican City has chosen themselves to not become part of the United Nations. 

So to answer your question, only Vatican City is by choice not a member of the United Nations. However it does enjoy full observer status.