The South Vietnamese city that became the site of a brutal 1968 Tet Offensive battle was Huế. The Battle of Huế began on January 31, 1968, when North Vietnamese and Viet Cong forces attacked the strategically important city during the Vietnamese New Year holiday.
Huế was politically and culturally significant: it had been the imperial capital of Vietnam and contained the ancient Citadel and Imperial Palace. Communist forces captured much of the city early in the offensive, while South Vietnamese and American troops held isolated positions and gradually fought to retake it in intense house-to-house combat.
The battle lasted until March 2, making it one of the longest and bloodiest engagements of the Tet Offensive. Allied forces recaptured the city, but much of Huế was severely damaged. Thousands of civilians were killed, including victims of executions and reprisals whose numbers remain disputed.
Saigon was also attacked during Tet, but it was not the location of this famous prolonged urban battle. Pleiku and Da Nang saw fighting during the war as well, yet Huế is the canonical answer to this question.