There were many major air operations conducted in Vietnam—I take it you mean US bombing operations against North Vietnam, rather than general air operations conducted in support of Allied troops in South Vietnam.
There are three major aerial bombing campaigns conducted by the United States against North Vietnam during the war:
Operation Rolling Thunder was primarily an attempt to boost South Vietnamese morale while also targeting key points of North Vietnamese infrastructure which was sending men and material down south.
Operation Linebacker was an attempt to halt the North Vietnamese Easter Offensive, which had seriously destabilized the ground situation for South Vietnam.
Operation Linebacker II was an attempt to force North Vietnam to the negotiating table after talks had stalled. This is also known as the "Christmas Bombings."
While the People's Republic of China had been sending military supplies and other aid to North Vietnam as early as 1955, beginning in July 1965, China began sending its own men to North Vietnam to bolster the North's anti-aircraft defenses in light of the devastating effect American bombing was having. As a result, many Chinese soldiers were manning AAA and SAM installations in North Vietnam. Between June 1965 and March 1968, there were over 320,000 PLA soldiers in North Vietnam, peaking in 1967 with approximately 170,000 PLA soldiers in-country.
Between 1965 and 1972 (the major periods of US bombing against North Vietnam), China sent 1,302,597 pieces of small arms, 42,126 pieces of artillery, and approximately 370 tanks to North Vietnam in military aid, among other supplies.
Interestingly enough, Russia initially waffled on full support for North Vietnam in the early stages of the war. However, by 1967, Russia had surpassed China as North Vietnam's primary supplier of arms and equipment. That year, 50% of the aid North Vietnam was receiving came from Russia. Additionally, Russia sent over 2,000 military advisers and technical experts to North Vietnam, and some of these men manned radar installations and even anti-aircraft installations.
For further reading, I recommend Ilya Gaiduk's The Soviet Union and the Vietnam War, and Qiang Zhai's China and the Vietnam wars, 1950–1975.