Short answers, pulled from the story.
Russia became a permanent member of the UN Security Council in 1945 when the Union of Soviet Socialist Republics was listed as one of five founding nations under Chapter V, Article 23 of the UN Charter. The arrangement granted veto authority to these great powers who had won World War II.
Eleven members of the Commonwealth of Independent States signed a declaration on the 21st of December 1991 agreeing that Russia should take over the USSR membership. President Boris Yeltsin personally took the Russian Federation's place at a Security Council meeting on the 31st of January 1992 following a letter sent to Secretary-General Javier Pérez de Cuéllar before Mikhail Gorbachev resigned.
International lawyer Yehuda Zvi Blum argued that Soviet membership should have lapsed upon the union's demise and required new admission for Russia. Professor Andrew MacLeod countered that Russia remained one of the largest states geographically and demographically after dissolution while noting the Vienna Convention on Succession of States did not enter into force until 1996.
Russia stated the veto was necessary for balanced and sustainable decisions during global discussions about UN reform. Global Policy Forum recorded statements from all five permanent members defending the current system even as Kofi Annan released his report In Larger Freedom in 2005 proposing more permanent seats.
Only five nations voted against the measure when the UN General Assembly held an emergency special session on the 2nd of March 2022. One hundred forty-one countries voted in favor of a resolution demanding a full withdrawal of Russian forces from Ukrainian territory while thirty-five abstained.