The United States had essentially been involved in Vietnam since World War II when the Office of Strategic Services sent both soldiers and equipment in the last days of WWII to help the Viet Minh.
But the conflict that Kennedy inherited from Dwight D. Eisenhower began in 1950 when the French Indochina War (1946-1954) got a completely new dimension. Up to that point, the French war in Indochina had been seen as a war of decolonization in which a nationalist organization was fighting to gain control of the country and gain independence. With the increasing military and logistical support from the newly formed People's Republic of China in '49-'50, the United States began to see the French war in Vietnam as less a war of decolonization and more of a war against communism. So as China saw the possibility to regain her traditional influence over Indochina by supporting the Viet Minh as well as to spread communism through armed force, the United States countered it by supporting the French war in Indochina in any way they could (except by actually getting involved militarily).
When the French withdrew from Vietnam after 1954 and South Vietnam came to be, the United States essentially took over the role that the French had played and began to support South Vietnam in the same way, this time actually sending boots to the ground in a limited capacity at first to train the newly formed Army of the Republic of Vietnam (ARVN). Gradually over the next coming years, the United States began to send an increasing amount of both military advisers and civilian personnel until reaching a milestone under the Kennedy administration when Green Berets where sent to Vietnam to carry out military and development missions.