While I'm not qualified to talk about the political situation in South Vietnam which ultimately may be most important, in military terms the Vietnam War is generally understood to have been lost because the United States didn't quite grasp or otherwise ignored the principles of counter insurgency warfare.

Firstly, in the 1960s the US military was structured to fight large scale, highly mobile armored warfare against the Soviet Union. The military was composed largely of conscripts in order to maintain high numbers. Units were structured and trained to maneuver and support objectives given at high echelons, typically at the division level or higher. Most intelligence assets were placed at the upper echelons as well, and intelligence was generally expected to flow from the top down, with higher headquarters responsible for collection and fusion of intelligence. This type of structure lends itself well to large scale maneuvers meant to envelope and destroy huge enemy formations in open warfare, but this type of fighting rarely happened in Vietnam. Instead, it made US units too slow to react to situations developing at much lower echelons than was expected.

Historically, successful counterinsurgencies such as the Malayan Emergency, or the tactics of the Selous Scouts in Rhodesia, utilized small teams of well trained soldiers, who also often acted as their own intelligence assets (intelligence flowing from the ground up). This strategy allowed these teams to infiltrate otherwise neutral villages to collect intelligence, and most importantly to separate the population from the insurgents. These smaller teams were then able to react much more quickly to developing situations than large units.

So basically, to sum things up, the US just wasn't prepared to fight a guerrilla war either in training or mindset. Decades of experience from World War 1 to Korea led to a military well trained and organized to fight World War 3, but which had completely forgotten lessons learned in low intensity wars like the Philippines. The US got bogged down and obsessed with "conventional goals" like overwhelming fire power and body counts, but these things were simply less important than protecting the population from insurgent infiltration.

I have more to say on this but I am on my phone, and it is exceedingly difficult to try to make a well thought out post typing with my thumbs. I will try to add more later.

EDIT: There is also the issue of the culture of military leadership in the United States during this period. Most of the top military leadership who were career soldiers were also veterans of other conflicts such as Korea or even World War 2, and this led to a noticeable preference for tactics such as large scale offensive sweeps which they were familiar with. Many of these leaders had been steeped in the conventional mindset for so long that they were unwilling to try anything outside of their experience.

Meanwhile the typical American conscript tended to be poorly motivated and generally unwilling to take risks. The US military suffered a lot of issues with discipline and professionalism in Vietnam which harmed its combat effectiveness.

There is also a hell of a lot more to this, but frankly there have been dozens of books written on the subject and I don't want to pretend like I'm even capable of scratching the surface with this one post.

Out of a military perspective, I've written a large post on this topic here.

One overlooked part of the failure of Vietnam was the underdog success of those in military power for the VC, namely Võ Nguyên Giáp, military commander during both the French and American invasions.

Within Giap, Warren argued that Giap's ability to overcome the Americans was heavily influenced by his understanding of the American people and the politics behind war in a foreign country. Giáp understood that a drawn out, brutal campaign would hurt the morale of not only the American soldiers, but the American citizens as well. The vicious and sporadic manner of guerrilla warfare was effective against both combatants and families alike, and coupled with Giáp's unwavering disregard for the life of his own soldiers, it left ample opportunity for battlefield horrors. The prospect of a loved one returning with missing limbs, altered psyches or potentially not returning at all weighed heavily on the citizens, considering their complete lack of understanding regarding the purpose of the war itself.

Aside from being considered a military genius, Giáp's understanding and abuse of the American people's influence over their respective politics was a considerable factor in the eventual retreat from Vietnam.

Source: Warren, James A. Giap: The General Who Defeated America in Vietnam. St. Martin's, 2013. Print.

Maybe I missed this scrolling down, but the documentary, "Fog of War" with former US Defense Secretary Robert McNamara, lists a lot of the reasons he thinks the war failed.

The most interesting one to me was that McNamara believed many Vietnamese thought the US was trying to follow the French and establish a colony in Vietnam.

He felt the US didn't successfully convey to a majority of the Vietnamese populace that this was not the case and opposition to re-colonization was something many could get behind.