There were many stages of escalation from countries accross the globe, leading up to US military involvement in Vietnam.
The events began with the French, who had returned to Vietnam to claim it for their empire after World War 2. It wasn't expected that this would take as long as it did. Incredibly the French were defeated by Ho Chi Mihn's Vietcong army at Dien Bien Phu. The Vietcong sought to promote communism in Vietnam.
Communism was feared in America, the largest capitalist country in the world. The American government felt if one single country became communist, it would eventually lead to its neighbouring countries becoming communist too. This theory, the invention of the US military theorists, was referred to as The Domino Effect.
Nearby to Vietnam, China, a country which was very large compared to its neighbours, had recently become a communist country, which could have had a massive effect if the Domino Theory was correct.
In Vietnam, it was decided that the country should be split into a North and South region, under The Geneva Convention. The Communists and Ho Chi Mihn would inhabit the North, and the capitalists lead by Diem would be based in the South.
When President Kennedy came to power in 1961, at the height of the Cold War, he was determined to stop the spread of Communism. Kennedy believed the Domino Theory and also believed that involving US with the affairs in Vietnam would demonstrate America's strength, hate of communism and power. He would send money, aid and advisers to South Vietnam.
Kennedy was assassinated in 1963, so it was never known what his real decision for military action may have been. Some modern opinions think Kennedy may have really wanted withdrawal as he could see the horrendous sequences of waging...