There are several power structures within the United Nations that allow it to function more efficiently and effectively. Your teacher may be asking for a specific six that they have gone over in class or from a text, but these are the ones I believe they would be referring to:
1) The General Assembly - Every member nation belongs to this, and has one vote in resolutions. Basically the Congress of the UN
2) The UN Security Council - The five members of the UN who have veto power and are on the Security Council forever,Britain, France, China, Russia and the US. Ten countries that serve two year terms on the 15 member Security Council. Every member country is eligible except the five permanent members.
3) International Court of Justice - Judicial branch of the UN, and responsible for cases involving international law and UN resolutions.
4) UN Secretary General - Currently South Korean Ban Ki Moon, they serve as the executive of the UN, and hold more power than ordinary ambassadors.
5) The Economic and Social Council - Charged with improving economic and social conditions in problem areas around the world.
6) The Trusteeship Council - Administers those countries and territories controlled by the UN, to oversee their best interests. All such territories have gained independence, though, so this council is currently inactive.
7) UN Peacekeeping Forces - An armed force that maintains the peace in various hot spots around the globe, with military force if necessary.