On March 24, 1999, the bombing of Kosovo began. And the stated purpose of this action was to stop the ethnic cleansing of Slobodan Milosevic's army against the Albanians in Kosovo. The debate in the media was limited to whether or not ground troops should be used; discussion over whether there should have been bombing to begin was limited.
In December of 1998 treaties between the Kosovo Liberation Army and Yugoslavia broke down. The following months were marked by military and civilian killings by both sides. In January 15 the Serbian military killed 45 Albanians, but this was the exception and not the rule. It is unlikely that this is what spurred the bombings.
International monitors were withdrawn for the preparation of bombing on March 22. Shortly after that, refugees started streaming out of Kosovo, having been forced out of their homes. At the time in media we heard of the genocide that occurred against the Albanians. In a retrospective in December 1999, the Wall Street Journal called the killings in Kosovo an "ethnic-cleansing light." Undoubtedly the killings and ethnic cleansing were terrible, but the great majority occurred after the bombing began. U.S. General Wesley Clark called this outcome "entirely predictable."
The bombings themselves also exacted a humanitarian toll: bridges were bombed during rush hour, cities known for their opposition to Milosevec were not spared.
Before the bombing began there were two peace proposals on the table: the Rambouillet Agreement suggested by NATO, and the other by the Serb National Assembly. The Rambouillet Agreement called for unrestricted access by NATO armies not only throughout Kosovo (of which it would have control over), but the rest of Yugoslavia as well. NATO would be immune to the laws of Yugoslavia, but NATO could give orders to Serbians. So, basically what it called for was surrender by Yugoslavia. The Serbian proposal called for equality and autonomy for Kosovo, ask for the UN for help on a peace proposal, but rejected a foreign military presence in Kosovo. It condemned the withdrawal of international monitors. The final proposal that ended the bombing rejected the heavy NATO presence throughout Yugoslavia, but Serbia agreed to have a military presence within Kosovo headed by the UN (it ended up being a NATO affair anyways.) The diplomatic options were not looked into enough by NATO, which is logical considering how stronger NATO is on the battlefield then the diplomatic one.
The Kosovo War was significant from a military standpoint in that it marked the first effective use of low technology local ground forces in combination with high technology air power provided by the United States. This combination would also prove effective in the United States campaign against the Taliban in Afghanistan in 2001.
During the early phase of the war, NATO air power had difficulty attacking Serbian ground forces which were well hidden and dug in. Faced with the unpalitible prospect of introducing NATO ground forces, NATO developed a new strategy. The Kosovo Liberation Army would attach Serbian ground forces making it impossible for those forces to stay under cover. Once the ground forces began to move, they were easy targets for NATO air strikes. Most Serbian losses occurred in the last week of the war, and these were sufficient to cause Slobodan Milosevic to decide to seek terms of peace.