Operation Linebacker II was the bombing offensive ordered by President Nixon of the United States against North Vietnam that took place December 18-29, 1972. The "main event" of Operation Linebacker II was a series of nighttime raids by large forces of U.S. Air Force B-52 bombers on targets in and around Hanoi, which these heavy bombers had not previously struck. In addition, there were daytime raids mostly conducted by U.S. Navy aircraft, and the B-52 raids were supported by large packages of other aircraft.
Militarily, Linebacker II was a American victory, although the North Vietnamese defenders did very well in the initial phases of the operation. The North Vietnam only effective defenses against the B-52s were surface-to-air missiles imported from the USSR. The B-52s flew too high for antiaircraft guns to be effective and Vietnamese People's Air Force's small force of night-capable manned interceptors was never able to challenge the USAF in the night sky. The B-52s, in addition to being formidable heavy bombers like their WWII ancestors, were a key part of America's strategic nuclear force and were no longer in production, so every B-52 that was lost was a permanent reduction in American military force and, potentially, a sign of strategic weakness.
On the first four nights of the operation (Dec. 18-20), the USAF's "maximum effort" attacks on Hanoi used unimaginative tactics that the DRV was able to counter and bring down a total of 12 B-52s. Concern over losses actually caused the USAF to cancel one wave of the December 20 mission and to withdraw its less-protected aircraft from the December 21 mission. The USAF then switched to less-defended targets for the next two nights (Dec. 22-23), suffering no losses, before unilaterally suspending the bombing for Christmas (Dec. 24-25). At that point, it may have looked like the North Vietnamese had won by driving off the American bombers. After Christmas, however, the USAF changed its tactics and was able to resume bombing Hanoi with only 4 bombers shot down in the final 5 nights of the operations (Dec. 26-29). The North Vietnamese defenses had largely been neutralized and the defenders were running out of missiles, with little ability to resupply them because of the damage caused by the bombing to the ports, facilities, and other infrastructure as well as attacks on the SAM batteries and radars. The B-52s were roaming the North Vietnamese skies with impunity. At this point, for political reasons, the bombing was ended and the operation was declared a success.
Although the USAF had selected military targets and wished, for many reasons, to avoid killing civilians, according to official North Vietnamese sources, 1,624 civilians were killed, with some estimates running about a thousand higher.
Politically, the result is not so clear cut. The ostensible purpose of Linebacker II was to get North Vietnam to accept peace terms acceptable to the United States, which at minimum would include return of American prisoners of war (mostly aircrew) and some language regarding Vietnam's political future that did not seem like a total capitulation. Facing a collapse in public support for the war, the United States had basically stopped bombing North Vietnam in 1968 (this ended the previous air campaign known as Rolling Thunder) and withdrawn ground troops from South Vietnam in 1971. In 1972, North Vietnamese launched a conventional invasion of South Vietnam. Initially successful, this invasion was slowed and then halted by a drastic escalation of the U.S. air effort including a resumption of bombing North Vietnam, in an air campaign known as Linebacker I.
Peace talks, which had been going on in Paris since 1968, resumed. In mid-October 1972, the United States and North Vietnam tentatively agreed to terms, but the president of South Vietnam objected because these terms did not include a requirement that North Vietnamese troops vacate South Vietnam--something Nixon did not seek and did not think could be achieved. The peace talks stalled and ground was lost, from Nixon's perspective. In early December 1972, the North Vietnamese tried to to drive a harder bargain, withdrawing concessions made October.
Nixon was under time pressure. Although he'd been reelected in a landslide in November 1972, the incoming Congress, which would take office in January 1973, was vehemently opposed to any continued involvement in Vietnam and seemed likely to force Nixon to suspend all military involvement and accept fairly minimal peace terms.
It was in that context that Nixon ordered Linebacker II: he wanted to bomb the North Vietnamese back to the negotiating table. This was somewhat successful. On December 27, 1972, the North Vietnamese indicated they wished to resume talks. A peace deal was signed on January 27, 1973 that almost exactly paralleled the terms reached in October 1972. For Nixon, this was a victory, since he got out of Vietnam before Congress forced his hand. Notably, the POWs started coming home. Thus, it's arguable--and this argument is particularly made by American aircrews and their supporters--that Nixon successfully bombed the North Vietnamese back to the negotiating table and brought our POWs home.
On the other hand, the terms Nixon got were simply what had previously been on offer, and they did not include guarantees for South Vietnam, which was militarily defeated and absorbed by North Vietnam within a few years with the United States sidelined. If Nixon bombed the North Vietnamese back to the negotiating table, he also bombed them into accepting favorable terms. Also, the main reason a peace deal was not made in October 1972 was that the South Vietnamese would not accept it, and much of Nixon's diplomatic effort in January 1973 was directed toward convincing the South Vietnamese president, Thieu, that the terms on offer were the best South Vietnam would ever get.
The North Vietnamese view has largely been that Linebacker II was "the Dien Bien Phu of the Skies," a costly but glorious victory that was essential to national liberation and unification. Given that North Vietnam won the war, it's hard to discount this position entirely.
Linebacker II was not very popular with the American and international press. Nixon did not make his goals clear in advance, probably because he wanted to limit the consequences of failure and remain free to ascribe to the operation whatever significance seemed appropriate later. So to many it seemed like Nixon was dragging the United States back into the war even though he was actually still looking for a way out. Although the civilian death and destruction were nowhere near WWII bombing levels, they attracted significant attention and condemnation at the time, including from western governments.
Your second question about whether the bombing could have been continued, presumably past December 29, 1972, is hard to answer. Militarily, sure: as explained above, the United States had defeated the North Vietnamese defenses. It is very hard to see how this situation would reverse. At some point, it probably would have been difficult to sustain high-intensity operations, but on the other hand the USAF was running out of targets suitable for B-52s to strike. On the other hand, politically, Nixon had run out the clock and it seems unlikely the new Congress would have permitted him to continue bombing. There is a good reason the bombing ended in calendar year 1972.
Best sources on Operation Linebacker II are two books by Marshall L. Michel III, who flew these missions as a USAF fighter pilot and later got a PhD in history. The Eleven Days of Christmas (2001) is a narrative history of the bombing that details how the USAF initially got it wrong, then corrected its errors. Operation Linebacker II 1972 (2018) is the rare Osprey illustrated book that one can recommend without qualification. It summarizes the tactical detail in Eleven Days but goes considerably further in some respects and gives a bit more context as well as provides more of the North Vietnamese perspective since Michel was able to collaborate a lot more with Vietnamese veterans and scholars after his earlier books got noticed. (Michel's first book, Clashes (1997), covered the earlier air campaigns over North Vietnam, Rolling Thunder and Linebacker I; he's also done a bunch more Osprey books.)