My 16 year nephew has to do a report on the Vietnam War. He asked me a question about it and I'm really unsure how to answer really. He asked me basically...
"How did the most powerful country in the world, with the best equipped, best lead, best trained military this world has ever seen lose the war?"
I was totally unsure how to answer really. Its kind of a daunting question...
It is daunting, and like many things, there probably isn't one (or even a few) correct answers.
First off, pre-Vietnam, for the most part, wars were fought against nations. That meant that you had specific goals to "win". Occupying a country that declared war was a good way to win, or inflict enough damage that the country surrenders. Either way, there were pretty clear-cut ways to "win".
In Vietnam, we were fighting an insurgency. How do you "win" against that? How do you even define "winning"? As you can see in the middle east, it is hard to eliminate insurgents. Plus, the nation you are defending really needs to help, and want to be free (for whatever definition of free you want). If the native population isn't going to help, it makes it even harder.
As to our military, if you read accounts of the time, we did not have the best military. Because of the dislike of Vietnam, politicians didn't want to activate the National Guard and Reserves to fight (all the older people in the those units didn't want to go, and they vote). So the military (mainly Army) had to draft large numbers of people that didn't want to fight. There were also mainly 1-year tours, which meant a lot of local knowledge had to be relearned. One Army office I believe was quoted as saying "We didn't fight one war for 12 years, we fought 12 1-year wars" (or something like that).
Add to the fact that the Army was prepared to fight the Soviets at the Fulda Gap, so didn't have training or tactics to fight unconventional forces in a jungle without the help of the local population in many cases. Then there were the officers that were only using Vietnam to advance their careers, and results be damned.
One example that got made into a movie, look at the movie and book about Hal Moore. He fought a 3-day battle in the middle of nowhere when they choppered in to look for the enemy. He "won", but after the battle was over, they pulled out. In a conventional war, you take and hold ground, and keep advancing to your objective. In Vietnam. there was no point to hold ground (for the most part), since there was no point. For the most part, the US Army never really adapted to this type of combat.
Then you have to deal with China and the USSR. It was a proxy war as others have said, so invading North Vietnam would have probably widened the war (just like in Korea, we couldn't invade China). And lets be honest. No one in the US cared about the country or the Vietnamese population, it was all about beating the USSR and score some points in the game they played.
Lots of reasons, to pick from, but the best lesson is to realize that you need a good plan and end-game for something like this. A more modern example of this was the first Gulf War. Bush 1 realized that invading Iraq would be a quagmire that wouldn't end so they just freed Kuwait. They wanted to be free, they would help, so it was a straight forward operation, with a clear goal, "Free Kuwait". The military went in, built up, and won. Then they went home (for the most part). That is a good plan.
The Second Gulf War is a great lesson to how to repeat Vietnam. We went in without any thought on how to "win", other then to kill Saddam, and decimated the country and infrastructure, and didn't know what to do once we had the country. And of course, Iraq is still screwed up to this day. In short, you better have a damned good reason to go to war, and a clearly defined (and attainable) victory and end-game, otherwise you get stuck in a never-ending war, which no one wants.