Like football, basically a game of two halves.
The first half (1946 - 1954) was the Vietnamese vs the French, ending in a rather predicatble win for the Vietnamese led by Ho Chi Minh and the Viet Minh, who had been trained by the US to fight the Japanese in WW2. A foreshadow of Afghanistan.
The second half was also a win for Ho Chi Minh.
The US Vietnam War
This one, as many people know, was a right mess, and marked the turning point in media coverage of modern warfare. The might of the USA was given a right good shoeing by a bunch of farmers with rusty old Type 56s, Lee Enfields and all manner of other US weapons given to them by (you'll never guess it) the USA.
Situation: Not Good
With communism threatening to spread through the world, the US was pretty keen to help people who were in charge of countries and who were fervently anti-communist, and it just so happened that a chap named Ngo Din Diem (funky lines and odd characters excluded) was a perfect candidate for such help in Vietnam. A lot of unrest and jungle punch ups later resulted in the creation of the National Liberation Front, also known as the Viet Cong, who rejected heavily the US advisers in the country.
Kennedy when he took over 1961 decided it was time to roll up the sleeves and get busy, and in true American fashion stepped in to provide a moral bastion to the peasantry of South Vietnam, who moved everyone that could possibly be uprooted into fortified camps under the Strategic Hamlet Program of 1961. Hearts and minds are key, after all.
Such was the popularity of Diem at this point that the CIA decided enough was enough, and whilst Kennedy remained happy-happy smiley face to the Catholic Vietnamese politician, they categorically stated they would not cease aid to a new power should there accidentally not on purpose completely by chance be a coup to overthrow the government.
In true Hollywood fashion a coup took place and Diem was whacked mafia style, on 2nd November 1963. Kennedy was pretty sad about this, but stopped worrying when he too was shot in the noggin' later that month.
I'll Cry When I'm Done Killing
Lyndon Johnson wasn't too fussed about Vietnam until the Gulf of Honking incident, in which a US intelligence gathering vessel angrily honked its horn at some North Vietnamese boats, who angrily honked in return. Somehow, perhaps due to overhonking, the Vietnamese vessels ended up re-enacting the Titanic's post-iceberg voyage. Everyone got all riled up, and the details were skirted over. Either way, North Vietnam was given a good spanking by the USAF's bombs.
With the Tonkin Resolution signed and given the go ahead, the president had the power to launch unrestricted military operations in South East Asia, except in Laos, who were neutral to the point that everyone including China and the Soviet Union signed a declaration stating they wouldn't operate in their territories. Shortly after, everyone did anyway. So good politics there.
1965 marked the point in which the US got down and dirty, commencing festivities with Operation Flaming Dart, Operation Arc Light and Operation Rolling Thunder. At the same time, the Viet Cong successfully kicked the balls of the USMC at Pleiku, and as they say, 'it's on'.
Things picked up big time, and more and more combat troops were stuck on ships and carted off to 'Nam. The first major batch was the USMC's initial delivery of 3,500 Marines. Later on, Mel Gibson decided to turn up and play too, bringing with him some guy from Buffy the Vampire Slayer, and his battalion from the 7th Cavalry rode on actual horses into the Ia Drang Valley and had a scrap with the North Vietnamese Army. After dropping napalm on their own blokes (fucking typical, one might say) and getting stuck right in they emerged victorious(ish) from their first pitched battle with the NVA.
Uh, A Little Help?
With the US quickly discovering the combined might of the Viet Cong and NVA was more than they wanted to handle alone, they sent out a Facey-B invite to their NATO and SEATO allies, most of whom told them to fuck off. Us lot, who weren't interested but didn't want it to seem like we wouldn't do anything at all, encouraged Oz to get involved, reassuring them that jungle warfare was jolly fun, and a right good crack.
The Empire Strikes Back
The Tet Offensive in 1968 was the biggest offensive of the war for the Viet Cong. Hidden all over the South, during the Tet holiday season when the yanks were all rat-arsed, they sprung the attack. A number of prominent cities were put to the torch, but the US, despite claiming negligence, were pretty savvy as to the chances of such an attack so fucked it off in short order. Not, however, without the loss of some 1,000 of their own, and 14,000 civilians for good measure. Charlie, on the other hand, lost some 32,000. No pain, no gain, as they say.
Notable battles include the Battle of Hue, in which the Americans levelled 80% of the city to expel the VC. If you want to see a good film including Hue, or Quang Tri (whichever it is), check out Full Metal Jacket.
Another notable moment in 1968 was the My Lai Massacre in which a bunch of US infantry, by this point fed up with the war, brassed up an entire town containing no enemy but lots of women and children. This is generally frowned upon.
Everyone, politicians, soldiers, the public, dogs, livestock, students, were well and truly ball-bustingly sick of Vietnam by this point, with the exception of a few (notably US LRRP teams). Everyone in the aforementioned categories had been shot up by the US Army (or National Guard) at some point in the 60s, and Nixon began the steady withdrawal of combat troops.
Too Much Effort Nixon's plan of Vietnamisation (they spelled it with a z, the retards, so I've corrected for the benefit of your eyes) meant that the ARVN (South Vietnam's own standing army) was given guns and plenty of bang to do the jobs themselves, as quite frankly, it was a lot of effort slapping Charlie about with increasingly more restrictive rules of engagement and fledgling public support for the war. Sound familiar?
The Ending and Aftermath
Paris Peace Accords '73
The Paris Peace Accords, after a lot of arguing, a bit of bombing, more arguing, some more arguing, a bit more bombing and some more arguing, were eventually signed. The US had a 60 day window to remove all combat troops from Vietnam, and went right ahead and did it.
Going It Alone
The South Vietnamese, after the departure of the Americans, utilised their remaining aid and had a few hefty battles with the Viet Cong. Unfortunately, the Viet Cong, free of the relentless US bombing, worked on the Ho Chi Minh Trail unimpeded and sent more and more stuff to ruin the day of the ARVN.
In 1974, North Vietnam declared the Paris Peace Accords no longer in effect and started a new campaign against the South. At the same time, the South was hit by a recession due to oil prices, and it all went tits up. Despite efforts to the contrary, the ARVN folded like origami and the communists ploughed their way to Saigon.
In 1975, the offensive reached Saigon. The USA made a hasty exit in the largest helicopter-borne evacuation to date, codenamed Operation Frequent Wind. Everyone, civilians included, tried to escape South Vietnam, and eventually it fell.
The Vietnam War wasn't very nice, but it has spawned a plethora of books, films and other inspired media, much of which is worth taking a look at.