It is difficult to argue that the United Nations was an effective organization with regard to the Rwandan genocide, as the mere use of the word "genocide" denotes great failure on the part of an organization founded in the ashes of World War II and the Holocaust.
The United Nations is comprised of two main organizations, the Security Council and the General Assembly. There are numerous smaller subordinate agencies, like UNESCO and UNICEF, but they have narrowly defined mandates and limited resources. The General Assembly provides a forum of all member nations to be heard. It can pass nonbinding resolutions on any issue in which a majority agree. It cannot, however, compel action. Its actions tend to be symbolic.
The Security Council, on the other hand, cannot necessarily compel action, but it can and does sanction action on the part of those member states willing and able to contribute. U.N. Security Council Resolutions represent the "law of the land," and carry a certain amount of legal, moral and political weight. Member states of the U.N. are obligated to comply with these resolutions, but the record of compliance can be inconsistent.
The weakness of the Security Council is inherent in the composition of its core members, the five major powers that emerged victorious from World War II. There are 15 countries represented on the Council, but those five, the United States, Russia (formerly the Soviet Union), China, Great Britain, and France, are permanent members of the Council and each have the authority to veto a resolution. Given the broad political divisions between these five countries, especially between North Atlantic Treaty members France, Great Britain and the United States on the one hand and Russia and China on the other hand, unanimity is a rare but valued occurrance.
What did this mean for Rwanda? It meant, as it did with the war in Bosnia-Herzegovina during the same time frame, inaction on the part of the only countries with the capabilities to act forcefully. Getting all five permanent members of the Security Council to agree on a resolution usually means finding the lowest common denominator. The watered-down language that often results from this dynamic can allow a bad situation to continue longer than would otherwise be the case. In the case of Rwanda, the Security Council passed U.N. Security Council Resolution (UNSCR) 918, which condemned the fighting and imposed an arms embargo, but otherwise did nothing to stop the massacres.
In conclusion, the United Nations failed to anticipate or act to stop a genocidal campaign waged by one ethnic group against another, a major failure on the part of an international organization founded for the express purpose of preventing such developments from occurring.