The Tet Offensive, which began a few hours after midnight on January 31, 1968 (the beginning of the Lunar New Year for the Vietnamese), is generally considered the event that turned the American population against the war. Even though the Tet Offensive resulted in about 33,000 dead Vietcong after two weeks of fighting, which was a staggering number of casualties for such a short period, there were also about 3,500 US and Allied troops killed, about 1,200 US dead, which absolutely shocked the public in the US. Just a few weeks earlier, General William Westmoreland, the commander of all US forces in Vietnam, told the American public that the war had reached a "turning point," that is, the US and its allies were winning. The Offensive, even though a military disaster for the Communist forces, proved that the US did not have control of the war.
Not long after the Tet Offensive ended, Walter Cronkite, a CBS news anchor and one of the most trusted men in journalism, visited Vietnam, and on his return announced to the American public that the US was "mired in a stalemate" and said that our only course is to negotiate to end the war and to leave "as honorable people." Cronkite, who was fondly called "Uncle Walt" by many American TV views, was incredibly influential, and his comments changed the public's perception of the war overnight.
At the same time, protests against the war by college students hit hundreds of campuses across the United States, and these protests were often violent and always disruptive. Many classes were cancelled during the worst of the protests and, in many cases, students who wanted to go to class found either no classes in session or discovered they couldn't get to their classes. These protests helped to convince the American public as a whole that the war was no longer worth fighting.
When Eugene McCarthy, a candidate whose goal was to end the war in Vietnam, had a very successful Democratic primary in New Hampshire, making it clear to both Democrats and Republicans that there was strong support among the electorate for ending the war, Lyndon Johnson announced that he would not accept the nomination for the 1968 Presidential race. Johnson's withdrawal could be considered, at least in the political life of the US, the point at which the US committed itself to getting out of the war and started looking for a candidate who could accomplish that.
The Tet Offensive of 1968, then, was the spark that ignited the desire among the American public to end the war, but the Offensive itself was just the spark. The American public, by this point, had seen hundreds of thousands of Americans sent to Vietnam and, by 1968, was tired of the cost in lives and dollars, tired of the constant civil turmoil (college protests and protest marches all over the country), and became convinced that our only rational course was to end the war as soon as possible.