What did the Tet Offensive show the American people?
The vulnerability of American ground forces in Vietnam.
American intelligence failed to pick up on a massive build-up of NVA troops around certain key cities, Hue in particular. Tet is the Vietnamese new year and Americans were expecting a quiet day. The attack caught the Americans entirely by surprise and immediately put into question America's ability to continue to wage that war.
Tet showed the American public the reality of warfare and an image in sharp contrast to that which they had been told by their own government.
The Tet offensive was launched by the communists during the Chinese New Year, Tet, in January 1968.
The communist strategy was to bring the fighting into the cities and creating the image if not reality that the communists were winning.
Prior to 1968, the American public had been told that aggressive search and destroy tactics by the American and South Vietnamese forces had pushed the communists right back. The view was that the war was being won.
When the American public saw the US embassy being occupied by the Vietcong, the savagery of he fighting in Hue, the impact of napalm, and the summary execution of a teenage Vietcong suspect, in the words of Clark Clifford "it looked as if the bottom had fallen out."
Militarily Tet was a very heavy defeat for the communists, but psychologically it was a crucial success. It turned an already sceptical American public even more against the war.