I would say it's biggest effect was in the media. The Vietnam War was the first to be broadcast into our living rooms nightly. The visions of causalities had a drastic impact on Americans view and opinions of the war.
I think the Tet offensive is important because it punctured so much of the optimism and so many of the hopes of Americans back home who had been told that things were going well and that the lives that had been lost were not in vain. Clearly the Tet offensive helped people back home to see the reality of the situation and that, after four, long and hard years of many deaths, the battle was definitely not close to being won.
The Tet Offensive was a nightmare for the Johnson Administration. It was also a moment where the American public was forced to embrace the idea that the war was unwinnable, their government had no clue on how to win it, and that the human cost is far too great to endure. When Johnson stood in front of the American public and proclaimed "there's light at the end of the tunnel," the Tet Offensive proved that he might not really know where the tunnel is. The coordinated attacks in both areas of densely populated concentration as well as in other areas around the nation was done with such force that the Americans were truly caught off guard. With the media being able to televise everything, Americans were able to see for themselves how bad bad could actually be. The Tet Offensive did much to sway public opinion against the war and against the President.
The above post is very accurate, and I have some additional thoughts. General William Westmoreland, the commander and public face of the Vietnam War on the evening news, had told us in early January 1968 that enemy attacks had dropped off to nothing, and that the "Light was at the end of the tunnel". There was talk about most troops coming home by Christmas of that year. Remember that by this time, the war was almost four years old for us, and had the complete faith of the vast majority of Americans.
Enemy attacks had dropped off to nothing because the Vietcong and North Vietnamese were preparing for the Tet Offensive. It took us completely by surprise, during a holiday cease fire. They attacked at every major US base in the entire country, the North Vietnamese Army rolled across the border and occupied the ancient city of Hue, and VC sappers even occupied part of the US Embassy in downtown Saigon, where it was supposedly safe.
Americans saw all of this on their television news at night. They felt duped. They felt like after four years of war and sacrifice, our leadership was unaware of the real situation. Walter Cronkite, the most influential man in TV news, gave an editorial saying the war was "unwinnable". President Johnson, upon hearing this, said "If I've lost Cronkite I've lost middle America!" He decided not to run for re-election.
For the first time, a majority of Americans said they were against the war, and for the next five years, the government never won the peoples' confidence back.
The Tet Offensive made the American public more pessimistic about the chance for victory in Vietnam and, just as importantly, made them trust their government less than they had.
Before the offensive, the US government had been assuring the people that victory was just around the corner. They claimed the enemy was all but destroyed. When the enemy attacked all across South Vietnam in this offensive, it proved that those statements were inaccurate.
This made Americans feel that the war was not being won after 4 years of huge troop presence in Vietnam. It also made them feel their government was lying to them and that reduced support for the war.
The Tet Offensive changed the perception that the US could win with some type of honor and leave Vietnam in a proud condition. That changed as soon as the reports of the offensive were reaching people's dining rooms through the TV.