Short answers, pulled from the story.
The Tet Offensive began in the early hours of the 30th of January 1968, when North Vietnamese and Viet Cong forces struck provincial capitals in II Corps and Da Nang. The main countrywide assault followed on the 31st of January, targeting more than 100 towns and cities simultaneously.
The North Vietnamese Politburo chose the Tet holiday as the launch date because the Lunar New Year festival period meant most ARVN personnel were on leave, reducing allied readiness. Hanoi had also announced a seven-day truce for the holiday, further lowering the allies' guard.
The Tet Offensive was a military defeat for North Vietnam. The popular uprising Hanoi anticipated never materialized, no ARVN units defected, and PAVN/VC forces suffered heavy casualties. The U.S. estimated approximately 45,000 PAVN/VC soldiers were killed in the first phase alone, and by the end of 1968 overall communist strength in South Vietnam had declined from roughly 287,000 to an estimated 251,000.
During the PAVN occupation of Huế, communist forces systematically rounded up South Vietnamese military personnel, government officials, civil servants, teachers, policemen, and religious figures. Mass graves discovered after the battle - the last uncovered in 1970 - contained victims who had been shot, clubbed, or buried alive. A captured VC document cited by historian Gunther Lewy stated that the communists had "eliminated 1,892 administrative personnel, 38 policemen, 790 tyrants."
Allied intelligence had collected mounting evidence of a major buildup, including truck traffic on the Ho Chi Minh trail rising from a monthly average of 480 to 6,315 by December 1967. However, intelligence methodology focused on enemy capabilities rather than intentions, and analysts concluded the communists lacked the capability for such an operation. One MACV analyst later said, "If we'd gotten the whole battle plan, it wouldn't have been believed."
The Tet Offensive shattered the Johnson administration's "success offensive," a months-long public campaign to convince Americans the war was being won. Public support for the war fell sharply, draft call increases intensified opposition, and the Johnson administration subsequently sought peace negotiations. The offensive also produced the photograph known as Saigon Execution, taken by Eddie Adams on the 1st of February 1968, which won the 1969 Pulitzer Prize and is widely credited with influencing U.S. public opinion against the war.