What is the 1961 united nations ban on nuclear arms?

I think you are refering to the 1963 Limited Nuclear Test Ban Treaty. On August 5, 1963, after more than eight years of difficult negotiations, the United States, the United Kingdom, and the Soviet Union signed the Limited Nuclear Test Ban Treaty.

The Treaty:

prohibits nuclear weapons tests or other nuclear explosions under water, in the atmosphere, or in outer space

allows underground nuclear tests as long as no radioactive debris falls outside the boundaries of the nation conducting the test

pledges signatories to work towards complete disarmament, an end to the armaments race, and an end to the contamination of the environment by radioactive substances.