It's quite a lot to know, and while it does fit (mostly) into the 20 year range, it's a little iffy. On that note, I'll do my best to explain the lead-up using some of the readings I've done for classes on the subject. Also, keep in mind that it's not called a genocide by most (if not all) authorities on the subject, and it's part of a wider conflict regarding the entire Yugoslav region (ie. Kosovo, Croatia, Serbia, etc., to name a few participants). Without further ado:

The Yugoslav Crisis in Bosnia was a crisis of nationalism, internal divisions within the Federation of Yugoslavia, and issues in political power in Bosnia. This is, of course, a very simple way of describing it. Nevertheless, it's important to note each of these factors.

Serbia, which had effective control of the federal level of the Yugoslav government (thanks to persuasive control over Montenegro, Kosovo, and Vojvodina), was also exacerbating tensions (as were all actors, to be fair). Slobodan Milosevic, the President of Serbia at the time, did a very good job of inciting nationalist fervor within Serbia by exaggerating and aggravating tensions between the Croats, Muslims, Serbs, etc. (various ethnic groups in Bosnia). In Bosnia, the Croatian-Bosnians, Muslim-Bosnians, and Serbian-Bosnians had the biggest sway, and they were facing a political crisis of sorts. The elections had created a split between them that prevented any one group from having a majority of power (each created their own political party based on ethnic lines, essentially), and the government had people from different parties at every level, only adding to the lack of unity. The Croatian-Bosnians and Muslim-Bosnians agreed to work together, in a sense, over the protests of the Serbian-Bosnians. They, led by Karadzic, essentially declared the independence of Serbian areas. With the support of Serbia, they moved to attack the Bosnians.

This, though, came after the Bosnians declared independence. This was a contentious point, and still hotly debated today; it's said that Germany and some other nations encouraged the declaration of independence too strongly, and helped create a huge issue. The federal government, controlled by the Serbians, was prepared (if memory serves), and declared the secession illegal. They had positioned Bosnian-Serb officers in the Federal Yugoslav Army (JNA) near the Bosnian areas, and used this advantage to immediately begin the ethnic cleansing of Muslim and Croat areas and villages. The JNA, and Bosnian-Serbs, initiated a program of mass expulsion of Bosnians who were not aligned with Republika Srpska (the newly established Bosnian-Serb "state" that followed Bosnian independence declarations). Houses, apartments, and other living centers were burned down. Rape, murder, and even some reports of internment camps of sorts came out of the area.

The international community, still, did little to nothing. The United Nations Security Council did pass an arms embargo, but they did so on the entire Yugoslav area. The Serbs, already having a higher advantage in firepower, now had more armories, more weapons, and a better-trained force at their backs. Croatia attempted to help the Bosnians, but though they had higher levels of manpower, they still faltered underneath the heavier firepower. The Serbians, who were the major aggressors according to most narratives, initiated a policy of ethnic cleansing that resulted in entire villages and towns being destroyed and emptied, as I mentioned before. However, this did not qualify as a genocide in the eyes of most. This is generally regarded as being because the genocide did not intend to actually "destroy" an ethnic group, as happened in Armenia (by most accounts), Rwanda, and Nazi Germany. It was merely to remove and kill as many civilians as necessary to make the other Bosnians surrender. Genocide, however, was typically regarded as an attempt in and of itself to destroy an ethnic group; not to make it surrender.

This may still have qualified as a genocide, in some eyes. However, it was not labelled such by the international community. This is typically regarded as being because of the idea of "never again". It was commonly said that if the United States, or anyone, labelled this a genocide, then they would be obligated to act to prevent it. Nuremberg had set this "never again" idea, and it was a legitimate fear that the United States would be drawn into the conflict (which they didn't want) if they labelled it as such (as other nations feared too). The Genocide Convention in 1948 also defined genocide, and obligated parties to act to impose penalties on those responsible for genocide, meaning that labelling it a genocide would be (most thought) sufficient to force intervention that no one seemed to want.

The UN force in the area, labelled UNPROFOR, was largely ineffective. They were ill-equipped with too small a mandate to really do anything. The Bosnians believed the international community would back them up if the Serbs got too aggressive after they declared independence, since they had given what amounted to a virtual guarantee of recognition upon declaration (once more, I believe this was primarily attributed to Germany pressuring other European nations to follow suit). On the other hand, Milosevic (after the gains by Serbian and JNA forces) hoped that calls for peace would let the UN be placed to enforce borders which would create a "status quo" situation of Serbian dominance and gains. Essentially he hoped that UN intervention in the form of peacekeeping (ie. not taking sides) would make sure that the ceasefire lines would remain the new borders, and lend credence to his hopes of Serbian gains in the area.

UNPROFOR was not established until February of 1992, and while they did accomplish some success in Croatia, this depended heavily on the local composition of the area. Areas the UN couldn't protect were cleansed harshly. Their mandate was really to protect aid, to keep some "safe areas" protected, and eventually to monitor a ceasefire in 1994

Eventually, however, NATO decided to intervene. This intervention was undertaken in 1995 (as far as full force goes), a full 3 years after most of the ethnic cleansing was estimated to have gone on. Over 100,000 were killed during this time, and 1,000,000 displaced, according to some estimates.

Until then, the international community established a no-fly zone in 1992, and really enforced it in 1993 (though NATO monitored some flights in 1992). In early 1994, NATO forces (in the first combat operation NATO had ever undertaken) shot down 4 Serbian jets. While NATO was tasked with helping UNPROFOR, cooperation and bombing really began in April, when NATO forces bombed strategic targets as requested by UN commanders. In 1995, an air operation called "Operation Deliberate Force" was undertaken, to strategically bomb Serbian positions (mostly described as retaliation for attacks on civilians). Eventually, the Dayton Agreement was signed, leading to peace in Bosnia. Bosnia & Herzegovina retained independence as a concrete state (though the federal government is not all-powerful), and joined into a federation between Bosnia & Herzegovina and Republika Srpska. This agreement was implemented by IFOR, an international Implementation Force tasked with transitioning everything from free elections to ceasefire agreements.

Hope that helps! I have more information, mountains, from the Silber and Little book I list below, but unfortunately I don't have it on me. Let me know if you have questions; I don't know about you, but I find this crisis immensely complex and mentally taxing to study, and I hope I haven't fudged any details. If I have, I'll be sure to go back and edit, and let you know :).

Sources:

Kuperman, Alan J. "Humanitarian Hazard: Revisiting Doctrines of Intervention." Harvard International Review 26.1 (2004): 64-68. Web.

Silber, Laura, and Allan Little. The Death of Yugoslavia. London: Penguin, 1995. Print.