The 1982-85 Lebanon War is, if you want to take a big-picture view of things, really a phase of the 1975-90 Lebanese Civil War. But as it involves the invasion of Lebanon by the Israeli army and therefore a substantial reordering of the conflict, it’s also perfectly reasonable to treat it as its own affair.
Which is not to say that the Israeli Defense Forces (IDF) hadn’t already been involved in what was happening inside Lebanon. From the beginning of the civil war in 1975 through its July 1981 ceasefire, the IDF supplied weapons and training to Christian forces like the South Lebanese Army and, above all, the Phalangist party. The IDF began to make regular incursions into south Lebanon (and, on occasion, as far north as Beirut), usually to lay mines or make airstrikes against PLO targets, and the PLO in turn would fire rockets and artillery into northern Israel. The PLO also continued its campaign of terrorist attacks inside Israel and in the Occupied Territories, as well as against Israeli targets in Europe.
The Israeli government of Menachem Begin accepted the PLO-Christian ceasefire of July 1981, but his national security team, under Defense Minister Ariel Sharon, was also planning out an invasion of southern Lebanon that was supposed to drive the PLO and Syria out of Lebanon entirely and magically turn the entire country Christian, or something like that. But they needed some kind of provocation to justify launching their invasion. A couple of events coincided to give them that justification. First, in late April 1982, an Israeli army officer was killed by a landmine while visiting a South Lebanese Army position. Why he was visiting a South Lebanese Army position, and who actually had laid the mine (it may very well have been the IDF) didn’t matter, and Israel resumed airstrikes against targets inside Lebanon. Then, on June 3, the Israeli ambassador to Britain was shot in an attack by the Abu Nidal Organization in London. Now, Abu Nidal had split from the PLO in 1974, and if he’d somehow been captured and handed over to the PLO in 1982 they probably would have executed him, but the Israelis didn’t care. They had already stipulated that any attacks against Israeli interests in Europe would be considered violations of the ceasefire, and now they had their justification to invade.
Israeli airstrikes began immediately, and the invasion followed on June 6. The IDF, using a combined air-land-sea plan, fairly quickly pushed as far north as the southern suburbs of Beirut. In the process, it steamrolled over whatever resistance it faced from Palestinian, Lebanese, and Syrian forces. Although the war continued technically until either June 1985, when the IDF pulled back into a belt of land across southern Lebanon that the Israelis designated as a “security zone” against further shelling from Lebanon into Israel, or June 2000 (if you really want to take a wider view of things), when the Israelis withdrew from Lebanon entirely, the heaviest action was over by September 1982.
It was at that point, following the Israeli occupation of West Beirut and the Phalangist massacre of thousands of Palestinian and Lebanese civilians at the Sabra and Shatila refugee camps (which is an atrocity best left for another time), that the PLO decided to evacuate Lebanon under the supervision of a peacekeeping force of US and European troops. Tunisia became the group’s new headquarters. Its status as Israel’s greatest enemy in Lebanon was quickly assumed by Hezbollah, which formed during the 1982-85 occupation as a paramilitary Shiʿa militia dedicated to driving the Israelis back out of the country and whose rise to prominence was actually aided by the fact that the IDF invasion had crippled the regular Lebanese armed forces. Hezbollah would be implicated in the 1983 bombing of the US Marines barracks in Beirut, an attack that caused the Reagan administration to prematurely withdraw from the multi-national peacekeeping force and, allegedly, got a young Saudi radical named Osama bin Laden thinking about how easy it was to hurt the Americans enough to get them to do what you wanted.
Overall the IDF lost 657 soldiers, compared with Lebanese casualties upwards of 20,000 dead (over 20,000 if the Sabra and Shatila casualties are included). Far from driving Syria out of Lebanon and strengthening the political position of Lebanese Christians, the Israeli invasion led directly to the Syrian occupation of most of Lebanon, and thereby left Lebanese Christians in a weaker position than they’d been originally.