The United States captured Guam on 21 June 1898 during the Spanish-American war. At that time Guam was officially a territory belonging to Spain, as it had been since the 17th century when it was taken over by Spanish colonists. On December 10 of the same year, Spain formally handed over the island to the US when it signed the Treaty of Paris. However, the treaty only came legally into effect the following year on 11 April 1899, so that could be seen as the year that the US formally took possession of the territory.
Guam was invaded by Japan during World War II and recaptured by American forces in 1944. After the war the people of Guam began to ask for more self-determination. Different laws have been passed since 1950, and today Guam has some autonomy although it is still officially an unincorporated US territory.
You can learn more about the history of Guam here.