I think the following infomation is accurate. Note: there have been several "Geneva" conventions. The US is a signatory, though with some exceptions as noted below.
The Geneva Conventions significantly predate the United Nations, which is why you often hear "the Geneva Convention" referred to in movies depicting World War II. The Red Cross (http://www.redcross.org/museum/pre1900a.html) tells the story of the original Treaty of Geneva (1864):After witnessing the suffering of the wounded following the Battle of Solferino, Henry Dunant (1828-1910), a Swiss businessman, initiated relief efforts that led to the founding of the International Red Cross and Red Crescent Movement.
Treaty of Geneva (1864)
Inspired by Henry Dunant's pamphlet A Memory of Solferino, and the resulting international feeling against the negligence that prevailed in the care of the war wounded, the First Geneva Convention convened in 1864 with 24 delegates representing 16 governments. The resulting Treaty of Geneva was signed by all but four delegates. Representatives of Great Britain, Saxony, Sweden, and the United States did not sign the treaty at that time.The United States signed the original Treaty of Geneva in 1882. You can find a more detailed history here (http://www.spartacus.schoolnet.co.uk/EUgeneva.htm).
The Treaty of Geneva has undergone three subsequent incarnations:The first convention was initiated by J.-H. Dunant; it established that medical facilities were not to be war targets, that hospitals should treat all wounded impartially, that civilians aiding the wounded should be protected, and that the Red Cross symbol should serve to identify those covered by the agreement. The second convention  amended and extended the first. The third  stated that prisoners of war should be treated humanely and that prison camps should be open to inspection by neutral countries. The 1949 conventions made further provisions for civilians falling into a belligerent's hands. Two 1977 amendments extended protection to guerrilla combatants; the U.S. did not sign them. Public opinion and disapprobation are the only sanctions that can be applied to violators.From The Britannica Concise (http://education.yahoo.com/search/be?lb=t&p=url%3Ag/geneva_conventions).
The Geneva Conventions in their modern form consist of not one but four conventions, adopted in 1949, plus the two protocols adopted in 1977: Convention (I) for the Amelioration of the Condition of the Wounded and Sick in Armed Forces in the Field. Geneva, 12 August 1949.
Convention (II) for the Amelioration of the Condition of Wounded, Sick and Shipwrecked Members of Armed Forces at Sea. Geneva, 12 August 1949.
Convention (III) relative to the Treatment of Prisoners of War. Geneva, 12 August 1949.
Convention (IV) relative to the Protection of Civilian Persons in Time of War. Geneva, 12 August 1949.
Protocol Additional to the Geneva Conventions of 12 August 1949, and relating to the Protection of Victims of International Armed Conflicts (Protocol I), 8 June 1977.
Protocol Additional to the Geneva Conventions of 12 August 1949, and relating to the Protection of Victims of Non-International Armed Conflicts (Protocol II), 8 June 1977.AFAIK, the United Nations did not play a formal institutional role in the Diplomatic Conference that proposed the 1949 conventions, and the United Nations does not play a formal institutional role in administering or enforcing them (other than the UN Secretariat registering the conventions and receiving information about ratifications, accessions, and denunciations--a general function for which the UN Charter (http://www.un.org/aboutun/charter/) (art. 102) provides). The formal administration of the conventions occurs through the Depositary of the Geneva Conventions and Protocols at the Swiss Federal Department of Foreign Affairs. But the impetus for the 1949 conventions came from the same internationalist sentiment that followed World War II and resulted in the United Nations being established, almost every United Nations member has ratified the conventions, and many sources--such as the BBC (http://news.bbc.co.uk/cbbcnews/hi/teachers/citizenship_11_14/subject_areas/united_nations/newsid_1773000/1773723.stm) and even the United Nations (http://www.hrweb.org/legal/undocs.html) itself--refer to the conventions as United Nations agreements. The "UN's ambivalent relation to the Geneva Conventions" is discussed by Roy Gutman in his article "United Nations and the Geneva Conventions" (http://www.crimesofwar.org/thebook/un-geneva.html).
Probably the best primary source for information about the conventions is the International Committee of the Red Cross (http://www.icrc.org/Web/Eng/siteeng0.nsf/iwpList74/77EA1BDEE20B4CCDC1256B6600595596).