Oh Jesus. That's nigh impossible to explain without giving you one part out of a twelve part story, but I'll give you the short version.
Basically Yugoslavia is actually like eight different countries. Serbia, Croatia, Bosnia, and then to a more dubious extent Macedonia. These countries have strong South-Slavic ethnic ties, and a history of having the shit beaten out of them by the Turks, the Austrians, Greeks, and everyone else nearby.
Yugoslavia is the idea that all of these should be one state, so that they can resist outsiders and be free. The problem is that because each state was itself a client of a larger power at some point, they all "smell" like that power to the others, so they don't trust each other at all. There are also religious concerns--particularly that the Serbs are Orthodox and the Croats are Catholic. The Bosnians are mostly Muslims, but that's actually less of a problem than the first part.
As a result, Yugoslavia only lasted as long as the figurehead who united it lived. Tito, when he died, the Croats immediately wanted to jump ship, and everyone else (except the Serbs) along with them. The Serbs were prepared to kill any number of people to keep Yugoslavia from breaking up, but that didn't go as planned, so now the region is fragmented and volatile.
The real reason this is a problem is that a lot of these people actually physically live in the other countries. There have been as many Serbs in Bosnia as there are Bosnians at times. As a result, each side feels it has a legitimate claim to everyone else's land.