Yes! Finally someone asks about my specialty (the Lebanese Civil War). Anyway, I'm going to have to preface this by saying that the conflict is an incredibly complex one, sides would change several times over the course of the war, and it is often impossible to guess the true motivations behind most of the major actors. I will be going over things in some (very) broad strokes, but I can elaborate on anything you may have questions about.

As for the regional situation, there are several important factors at play:

The first is the situation of Palestinian refugees, and the Palestinian Liberation Organization (PLO), which was an umbrella organization for the various Palestinian militant groups created in 1964. After the 1948 War and the creation of Israel, many Palestinians fled, or were forced to flee, Palestine into neighboring Jordan and Lebanon. In both cases they were not given citizenship, with the expectation that they would eventually be allowed to return to their homes. After the devastating Arab loss in 1967, the Palestinian community generally lost faith in the Arab to press their cause against the Israelis, and the PLO became a much more militant group. The PLO was mostly based out of Jordan, and would regularly launch attacks into Israel, with the Israelis retaliating against them in Jordanian territory. The PLO would also begin to create a state within a state in Jordan, and by 1970 there was a very real fear that the Jordanian Hashemite kingdom would be overthrown by the PLO. This precipitated the events of "Black September", when the Jordanian Army would go to war the PLO, and would eventually force them to relocate in Lebanon, where they would continue to launch attacks against Northern Israel.

The second important movement was the Pan-Arab movement, led by Egypt's charismatic strongman Gamal Abd El Nasser. While the 1967 war, the failure of the United Arab Republic, and Nassers death would greatly diminish the Arab Nationalist cause, it still had a very powerful meaning to many Arabs across the region at a time, and Arab unification in some form remained the goal for many political groups. Among them, at least in name, was Hafez Al-Assad's Syrian Baath party, which made no secret of its desire to in some form reunify with Lebanon (or at least make it into a client state), which had traditionally been a part of "Greater Syria".

The situation in Lebanon was, even before the PLO's arrival in force, a very tenuous one. Lebanon is an extremely diverse country, with 18 recognized religious groups. The 1948 Lebanese National Pact, which was the agreement the Lebanese governmental system was founded on, created a confessional system which guaranteed proportional representation to each confessional group. In reality, the last census held in Lebanon was in 1932, and it did not necessarily reflect the reality on the ground. The split in parliament was a 6:5 Ratio favoring Christians; in addition, the President was to be a Marionite Christian, the Prime Minister a Sunni Muslim, and the Speaker of the House a Shiite Muslim. The system also made the executive branch the most powerful branch of government; this essentially meant that Lebanon was a Christian, specifically Marionite, dominated country.

The National Pact also specified the national character of Lebanon. In 1948, the Marionite Christian community, as well as several other communities, were extremely concerned that following decolonization they would be made into a part of Syria, and thus become a minority in a Sunni dominated state. Many Lebanese Sunnis however deeply wanted to be reunified with Syria, and were early Arab Nationalists. There was also a good deal of contention on the orientation of Lebanon, whether towards the western world, specifically France, or towards the Arab World. The National Pact would thus be a major compromise to all communities, and would (simply) describe Lebanon as "an independent country (so not part of Syria, and also not under French Dominance), which had an Arab face (so the West could not become too involved in Lebanese affairs).

This was, and is, an inherently unstable system. There were several "near civil wars" in between 1948 and 1975, especially in the crisis of 1958 which would cause the United States to intervene. However, the arrival of the PLO would be a major catalyst towards the war. The PLO would continuously launch attacks into Israel and maintain its arms despite the existence of the Lebanese Army. Israel would in turn retaliate into southern Lebanon, as they had done in Jordan, frequently killing Lebanese in the process.

The Lebanese would become divided based on their position on the Palestinian issue. Most (certainly not all) Marionite Christians saw the Palestinians as a major existential threat to the sovereignty of Lebanon and the Lebanese system; many also saw the PLO and their supporters as a serious leftist and communist threat. Most (again, not all) Sunnis supported the Palestinian cause against Israel. This was also fueled by a desire to change the system within Lebanon and Arab Nationalist sentiment. Tensions would rise between two major blocs: The Lebanese National Movement, led by the Druze leader Kamal Jumblatt (generally Sunni and Druze, sympathetic to the Palestinian cause, Arab Nationalist and leftist) and the Lebanese Forces, and the Lebanese Front led by Camille Chamoun (generally Christian, right wing, status quo, anti-Palestinian, and western oriented). The war would semi officially break out in 1975, with a series of clashes between the groups eventually escalating to an attack on a Christian church, and a retaliatory massacre against a bus full of Palestinians in Ain El-Remmaneh, a Beirut neighborhood.

The war would spiral out of control from there until June of 1976, when a Syrian force would "intervene" in Lebanon, on the side of the Christians who had invited them into the country. The Christian (the Marionite estiblishment in the form of then President Franjieh, Camille Chamoun, and the Gemyael family, as well as some of the conservative Sunni population) rationale was that they needed the Syrians protection, and the Syrian rationale was that it was a means of fighting anti-Syrian Islamists who had fled to Lebanon and were fighting with the National Movement, as well as to reclaim some control over the Lebanese state. This would bring a temporary peace, however, the PLO would continue to Launch attacks against Israel in the south. The Israelis, in retaliation for the continued PLO attacks and a fear of the Syrian presence in Lebanon, would launch Operation Litani and invade South Lebanon in 1978 in an attempt to drive out the PLO.

Here is where the UN intervention you asked about comes into play. The Israelis would invade up until the Litani river and create a buffer zone in South Lebanon against PLO attacks. This panicked international powers, especially the US and the USSR, who feared that the Israelis or Syrians could begin fighting each other, effectively creating another conflict between the superpowers. The UN security council would thus pass resolutions 425 and 426, which would set up the UN Interim Force in Lebanon (UNIFIL), and place an interim force in South Lebanon to ensure the Israeli withdrawal and a demilitarized South Lebanon. Israel would effectively withdrawal in 1978, but would maintain some troops in a 12 mile "security zone", as well as a client Lebanese militia called the South Lebanon Army (SLA) to ensure Israeli interests in the south.

As to how the UN impacted the situation? Well, to be honest they more or less didn't. UNIFIL is generally considered to be a major UN intervention failure. They (UNIFIL) were regularly attacked by both the SLA and the PLO with little retaliatory power. The war in South Lebanon between the SLA and the PLO would continue for much of the Lebanese Civil war. Israel would invade Lebanon again, in 1982, again under the pretext of PLO attacks, and this time would occupy the whole South and siege Beirut. The later emergence of Hezbollah would only complicate this. The Lebanese Civil war would end in 1990; Israel would not withdraw from its positions south of the Litani until 2000. All of this will be breaking the 20 year rule but- Israel would invade Lebanon again in 2006, Syria would remain in Lebanon until 2005, and UNIFIL is still in Lebanon, and shows no signs of leaving.

Sorry about the length, but like I said, it's a very complex conflict with a lot of factors and payers and shifting alliances, and this is really only the lead up to it. To read more, I would suggest a few books here: Fawwaz Traboulsi's "A History of Modern Lebanon" is one of the best (and honestly one of the only) impartial Lebanese histories of modern Lebanese history and the Civil War. Robert Fisk's "Pity the Nation" is an excellent source on the war itself, though like all Fisk books it can be a bit self aggrandizing. Both David Hirsts "The Gun and the Olive Branch" and Thomas Friedman's "Beirut to Jerusalem" are problematic, but they are still generally accurate and have good information. Farid El-Khazen's The Breakdown of the State in Lebanon: 1967-1976 is a great source for the "pre-conflict" stage, although it certainly has its biases. If youre looking for a specific topic, person, militia, or issue in the war, let me know, and I can point you in the direction of some good academic sources on it. And again, please remember that this is a very very very broad overview of the causes, regional dynamics, and early stages of the conflict. I'd be happy to provide any clarifications or answer any questions.

Edits: Formatting and spelling, because I'm just garbage at it.