Which United Nations body is the only one that allows some of its members to have veto power?

The United Nations Security Council

Explanation:

The United Nations Security Council made up of five permanent members such as France, China, the United Kingdom, the United States, and Russia have the power of veto which is exclusively exercised by them on any "substantive" resolution.

The Veto power perhaps, is the most visible difference between permanent and non-permanent members of the UN. Permanent members utilize the power of veto to defend their national interests.