The Vietnam war started as a war for independence from France by the Vietnamese people, the resistance that they formed was called the Vietminh. You can see the Vietminh as a precursor to the Vietcong, but with the twist that the Vietminh just fought the nationalist cause of the liberation of all of Vietnam. The French eventually, despite a lot of effort lost that war. This insurgency ended with the Geneva Peace accords which among other elements decided that:

-An immediate Ceasefire -The separation of Vietnam into North and South Vietnam in which the Vietminh controls North Vietnam and the French control South Vietnam

Similar accords where signed about Laos and Cambodja, other French Colonial possessions, however none of these accords where signed by the US, because the Peoples Republic of China had signed it too, and the US didn’t want to acknowledge them as a legitimate state. (The US government still considered Taiwan, or the Republic of China, as the legitimate Chinese government at this point) In South Vietnam the King of Vietnam was the supposed head of state but he was deposed by his Prime Minister Ngo Dinh Diem, who broke with the French and established South Vietnam as de facto independent. The US supported this coup because Diem, as the US, opposed the Geneva peace accords. This opposition to the peace accords meant that Diem also didn’t allow the elections of 1956 to occur, which likely would’ve been won by Ho Chi Minh, the leader of North Vietnam and former head of the Vietminh. Diem’s refusal ignited a lot of protests against him due to the popularity of Ho Chi Minh, even among the Soldiers many of whom where former Vietminh guerillas who had supported the peace accords because they believed that in time Ho Chi Minh would rule a united Vietnam. This protest called itself the National Liberation Front of Vietnam (NLF) and was called by Diem the Vietcong.

In the meanwhile America became more interested in this national conflict due to the strategic importance of Vietnam and the threat of it becoming aligned with the Communist block. As such the US starts to intervene into the problems in South Vietnam and helps the Vietnamese army to quell the protests, which had evolved into an armed revolt by the Vietcong. Slowly the US got more involved under Lyndon B Johnson and Nixon. Especially the Gulf of Tonkin incident, the alleged attack of the North Vietnamese navy on an American Destroyer gave the US an excuse to expand US support for South Vietnam. By this time the North Vietnamese army was also engaged in hostilities against the South Vietnamese and American troops. The Vietcong, though fighting in the South where supplied by the North Vietnamese government trough the Ho Chi Minh Trail that was a ‘road’ leading from North Vietnam over the border into Cambodia into South Vietnam. This road was also bombarded by American Aircraft. The American kept escalating the conflict, trying to resolve it militarily but after the Tet Offensive, a combined offensive by the Regular North Vietnamese army and the Vietcong, in 1968 the public support for the War dwindled in America. This happened because this proved to the American Public at least that the War was far from won. The war eventually lasted till 1975, when Saigon was taken by the North Vietnamese Army. By this time the US had retired all it's forces from the conflict.

This in short describes the origins of the conflict in Vietnam, and as per your question the main parties involved. TLDR: The Vietcong were North Vietnamese sympathizers fighting in South Vietnam for a United Vietnam. The South Vietnamese where initially a French territory, and after the 1955 coup an independent country that was supported by the US. The North Vietnamese where the independent country of Vietnam after the Geneva Accords, but wanted to reunite the country under the leadership of Ho Chi Minh, the former leader of the Vietminh but current head of the North Vietnamese state, with the North Vietnamese army as their forces. And lastly the American army which supported the South Vietnamese against North Vietnamese attempts to unite the country by force, which eventually happened in 1975 after the US stopped supporting the South Vietnamese.

Sources:

• KERREMANS, B. and LAENEN R., International Politics since 1945, Leuven, 1999.

• SCHULZINGER, R.D. A time for War. The United States and Vietnam, 1941-1975, New York, 1997.