The Tet Offensive, named for the Vietnamese New Year on which the North Vietnamese Army and its Viet Cong guerrilla allies launched a surprise large-scale series of terrorist attacks against the government of South Vietnam and the United States military and diplomatic presence, marked the beginning of the end of the U.S. role in Southeast Asia.  Until that January 30, 1968 offensive, the administration of President Lyndon Johnson and his military commander in South Vietnam, General William Westmoreland, had been able to argue persuasively to the American public that the United States and its South Vietnamese allies were prevailing over their communist enemies.  The Tet Offensive, while an overwhelming military victory for the U.S. and the Army of the Republic of Vietnam (ARVN) -- the Viet Cong and its once-formidable infrastructure was largely eliminated as a major problem and the North Vietnamese Army suffered serious losses in battles like that at the ancient city of Khe San -- the public perceptions back home in America were that the United States suffered a major setback during the surprise attack, the exact opposite of what actually transpired.  Walter Cronkite, the widely-respected anchorman for CBS News, declared on his nightly broadcast that the Tet Offensive represented a huge setback and that the Johnson Administration's projections of victory were fallacious.  Cronkite's reputation for integrity carried great weight with the public, and public perceptions of the war in Vietnam were turned decidedly against the U.S. role there.  

In short, the Tet Offensive marked the end of much of the American public's support for the war in Vietnam despite the resounding defeat during that offensive suffered by the communists.  The perceptions were more powerful than the facts, and subsequent revelations (mainly by the leaking of the Pentagon's own in-house history of the war, The Pentagon Papers, to the media by Daniel Ellsberg) that Johnson and others in his administration had been less than truthful in describing the situation in Southeast Asia eliminated support for his administration, causing him to retire from politics rather than run for reelection.  Support among many Americans for the war effort would never be regained.

The Tet Offensive is generally seen as one of the most important single events in the war.  It affected the course of the war by badly degrading the will of the American people to fight that war.

In 1968, the US had been involved in the war for over a decade.  It had been heavily involved, with hundreds of thousands of personnel in South Vietnam, for about three years.  Americans were becoming restless.  The Johnson administration tried very hard to reassure the American public that all was well.  They argued that the enemy was on its last legs and that the war would be over soon.  When the communists launched the Tet Offensive, they destroyed the credibility of the Johnson administration.  Even though their offensive failed in military terms, it was strategically very successful.  It showed that they were not really all that close to being defeated.  That made it clear that the government had not been truthful with its people.  The American people lost faith in their government and started to push for an end to the war.

In this way, the Tet Offensive played a very important role in changing the course of the war.