What happened in Vietnam before the Vietnam War regarding the French? How did the U.S. get involved?

I'm assuming you just want an outline of what happened, since you asked such a broad question. Once you've digested that you might have some more specific questions.

Nguyễn Dynasty

European powers had been trading in and sending missionaries to the area comprising modern-day Vietnam since the 16th century. French people first came into armed conflict in the region in 1787 when the Catholic priest Pigneau de Béhaine organized volunteers to help the Nguyễn Dynasty, which was the last ruling family in Vietnam before the French.

Throughout the first half of the 19th century, French forces were moved to the country in increasing numbers, ostensibly to protect the Catholic missions there, which began to worry the Nguyễn. Unable to prevent the Nguyễn from attempting to expel Catholic missionaries through diplomatic means, Napoleon III sent a naval invasion in 1858, capturing present-day Da Nang for a time and then capturing and securing Saigon within a few months. At this point the French commander had orders to secure a treaty ensuring the safety of Catholic missionaries and to not pursue any further territorial gains due to the Second Opium War drawing away French forces. During 1860 and 1861, Nguyễn forces attempted unsuccessfully to recapture Saigon, and the French defending forces were heavily reinforced after the Second Opium War ended.

At which point France went on the offensive again and over the course of several campaigns from 1862 - 1885 they took over all of modern-day Vietnam and most of Cambodia. They also had to fight off repeated attempts by the Chinese to reassert control, as they saw themselves as the traditional overlords of Vietnam and resented the growing French influence in the region. This culminated in the Chinese loss in the Sino-French War. After their victories, France named this new, consolidated colonial possession French Indochina.

French Indochina

From 1885-1895, an unsuccessful rebellion against France was led by Phan Đình Phùng. France did not experience any major organized resistance here for some time, and they took advantage of this relative peace to expand their territories in the region.

French territorial ambitions led to the Franco-Siamese war in 1893. France used border disputes and the Paknam naval incident to provoke hostilities with Siam (based in modern-day Thailand), which was forced to give up most of Laos to France (and a region of Burma to the British). In 1906 France succeeded in manufacturing another casus belli against Siam, and this time they took the rest of Laos and Cambodia and some parts of Thailand.

Vietnamese nationalism continued to grow during, before, and after the first World War, culminating in a 1930 rebellion of Vietnamese army units in the Yên Bái mutiny. This was encouraged by the Vietnamese Nationalist Party, Việt Nam Quốc Dân Đảng (VNQDĐ). They had previously relegated their activities to clandestine sabotage and the like, but felt pressured by French crackdowns to do something that might inspire the general populace to join the fight and shake off the imperialistic yoke. The majority of Vietnamese soldiers refused to take part, however, the general populace did not rise up, and the French response was quick and brutal. All the main VNQDĐ leaders were arrested and swiftly executed.

This ended the military threat for the time being, but France was a bit shaken by the event and made sweeping changes to the organization of their colonial troops in response. They drastically reduced the number of ethnic Vietnamese in the armed colonial forces, a reduction that was replaced with ethnic French troops and ethnic minorities from other French colonial possessions.

World War II

Everything was generally pretty quiet until World War II. Japan invaded French Indochina before France fell to Germany in 1940 and took parts of Vietnam. Once France fell to Germany, the German-occupied Vichy French regime granted access to much of Vietnam to Japanese forces.

Thailand took advantage of the situation to initiate hostilities in the French-Thai War, which lasted from 1940-1941. Thai land forces performed well, but gained back only small amounts of territory before French naval forces gave them a crushing defeat. It was at this point that the Japanese intervened and forced the French to give back the territory it had taken from Thailand.

While the Japanese occupied Vietnam, the Viet Minh formed under Ho Chi Minh to resist the Japanese. This group was heavily supported and funded by the US.

In 1945, once Germany had been defeated and France had been liberated, there was a unique situation where the Vichy Regime was no longer in power in France, but their colonial administration still controlled Indochina. Japan, worried that the Vichy French colonial officials would begin taking orders from the re-instated French government to fight them, decided to take full control of Indochina. They persuaded the Lao crown prince, Savang Vatthana, to declare Laos independent, and then they started the Second French Indochina Campaign, which was essentially a coup. All French forces that resisted were massacred or executed. They did not hold power for long, however, as they relinquished it once news of Japan's surrender reached them.

First Indochina War

FDR and his generals privately made it clear to France that they were NOT to attempt to reacquire their possessions in Indochina after the war. France was embarrassed by their quick defeat in WWII, and wanted to quickly re-assert itself on the world stage. The US offered Indochina to Chiang Kai Shek to administer, but he refused the offer. He did, however, invade northern Vietname with 200,000 troops to accept the surrender of the Japanese forces there, and then threatened war against France if they did not stop maneuvering against Ho Chi Minh's forces. Chiang Kai Shek forced France to give up all their Chinese possessions in return for the withdrawal of Chinese forces, allowing France to re-occupy the area.

During this same time period, Ho Chi Minh convinced the last surviving member of the Nguyễn Dynasty, Emperor Bảo Đại, to abdicate so Ho Chi Minh could become the first president of the Democratic Republic of Vietnam. French forces, augmented by British and Japanese forces, sparked the French Indochina War in an attempt to re-establish control.

From 1946 until 1954, France fought the communist forces. The US sent financial, naval, and air force assistance to French forces in Indochina from 1951/2-1954, and conducted covert CIA actions in the last year of the war. The war became increasingly unpopular with the French people, and with the rest of the world in general. Other colonial powers had been giving up their former possessions relatively peacefully, but the French continued to hold on. Other powers caused calamity through callousness in the drawing of borders and the preparations for transition, but France was causing tremendous casualties by fighting bitterly in a vain attempt to reclaim their former glory.

Partition of Vietnam

Following the surrender of French forces at the Battle of Dien Bien Phu, France agreed to hold the 1954 Geneva Conference. In it, the communist Vietnamese delegates successfully lobbied for a division of Vietname into North (Democratic Republic of Vietnam) and South (Republic of Vietnam), against the strenuous objections of the non-communist Vietnamese delegation. The expectation was that there would soon be local elections to reunify the country. The American Plan, which was rejected following opposition by the USSR, was to have reunification elections conducted under the supervision of the UN.

Emperor Bảo Đại, from his home in France, appointed Ngô Đình Diệm as Prime Minister of the Republic of Vietnam, while Ho Chi Minh had already been declared President of the Democratic Republic of Vietnam. With the support of the US, Ngô Đình Diệm used a referendum to remove Bảo Đại from any position of power and establish himself as the sole leader in the south.

Fearing the possibility of sharing or losing power, Ngô Đình Diệm prevented the reunification elections from occurring, which caused Ho Chi Minh to activate Viet Minh cells that had been left behind in the south. From 1954-1960 an insurgency raged in the south, although the extent to which this was actually controlled by the north is debated. It wasn't until 1960 that the Viet Cong were created, with the goal of uniting all groups that opposed the south's government, which included non-communists.

US Involvement

As the situation worsened, the US began to get more and more worried about the encroachment of communism and slowly ramped up the assistance that had been in the region assisting first the French, and then the Republic of Vietnam, since 1951. For this reason, the exact beginning of the hostilities between communist Vietnamese forces and the US is a bit fuzzy, but by 1964, after the Gulf of Tonkin incident, the US was fully committed to the war that would follow ...