The Vietnam War was part of the larger Cold War. The Cold War, in case you don't know, was a rivalry between the capitalist United States and the communist Soviet Union.

For many years, Vietnam was part of a French colony. After World War II, the Viet Minh (a communist-flavored independence movement led by Ho Chi Minh) fought the French for Vietnamese independence. Due to the emerging Cold War framework, the U.S. assumed the Viet Minh were Soviet puppets and began supporting the French with money and arms. In 1954, the French lost and Vietnam was divided between north and south. At the end of this war, North Vietnam was a communist dictatorship led by Ho Chi Minh while South Vietnam was all that remained of French colonial rule. With the departure of the French, South Vietnam was left as an anti-communist dictatorship with U.S. backing.

After the war, there was supposed to be an election which would reunite the country under one government. However, the U.S. and South Vietnam prevented the elections, claiming that they wouldn't be free elections in North Vietnam. Eventually, South Vietnam was harshly cracking down on dissent and the Viet Cong, a communist guerrilla army, was invading the south with much support from the rural peasants. Which of those two events happened first is a matter of heated dispute.

The U.S. supported the South Vietnamese government's efforts to put down the Viet Cong and gradually became more and more involved in the war. This culminated with the Gulf of Tonkin incident, when a couple U.S. ships were (allegedly, at lest) fired upon by North Vietnamese ships. This provided the pretext to send U.S. troops outright.