Chechnya is a separate thing. Chechnya is a nationalist conflict between the Chechens and the Russians, who are occupying Chechnya (or the Chechens are better off with Russia and should stop complaining, depending on whose side you're on I guess). This is over in the Caucases, across the black sea and several countries from the Adriatic side of the Balkan peninsula, where Yugoslavia was.

So in a nutshell. In former Yugoslavia (established as a socialist country not aligned with the USSR after WWII) there were three major ethnicities; Croats, Serbs, and Bosniaks. Their languages are mutually intelligible and used to be known as Serbo-Croation, but now they are called BCS (Bosnian, Croation, and Serbian). In addition to linguistic ties, ethnically there is not much separating them. It is mostly an issue of culture and religion and which empires historically held sway in the different regions. Broadly speaking, "Croatia" is in the west, with a coastal part encircling the other countries "Bosnia" in the middle, and "Serbia" is in the east (basic geography fail on my part, now corrected). But you have to understand that all of the ethnic populations were pretty mixed.

Croats are mostly Catholic, generally use Latin script, and historically were under the influence of the Austro-Hungarians and more connected to the general post-Latin world. The Serbs are Orthodox, mostly use Cyrillic script, and have the biggest historic connection to the Russian empire, and historically the most support for pan-Slavism. Bosniaks are Muslim, and they are historically connected to the Ottoman Empire (even though they also generally use Latin script). So while religion certainly played a role in this conflict, ethnicity/culture is perhaps the defining thing. It is also worth noting that early on in the 20th century the Bosniaks were not really recognized as a separate ethnicity

Some of the later conflict breaks the 20 year rule. But the basic gist of the beginning of the conflict was that many Serb nationalists "saw the writing on the wall" - or at least thought they did - with the end of the Cold War and collapse of the USSR, they thought that socialism and Yugoslavia were both on the way out. It's impossible for me to say if Yugoslavia definitely had to go, but it seemed to not be able to outlive Tito, the cold war dictator, for long. Many elements of the JNA, the Yugoslavian Army, declared themselves pro-Serb and began flying the Serbian flag instead of the Yugoslavian flag. Obviously, this alienated any Croat or Bosniak soldiers in these units and many of them deserted, and when things started getting violent they formed their own militias (troublingly, in Croatia, some of the militias used the same names as fascist militas during WWII had). At the beginning of the conflict, the Serbians had access to most of the weapons of the former Yugoslavian army, an obvious advantage when beginning a civil war.

Now, remember what I said about the three ethnicities being pretty mixed, geographically speaking? While it's true what I said that the Croats were mostly in the northwest, the Serbs and the Bosniaks were essentially all tied up. In the area around Sarajevo, you couldn't say there was a clear majority, although I believe there were probably more Serbs in total than Bosniaks. The war was mostly fought here, between Serbian and Bosnian militias, with the Serbians main war goal to seize as much land as possible and claim it was ethnically Serbian. This is where things begin to descend into really hellish levels of violence against civilians - on both sides, as the Bosniaks began retributive strikes. The siege of Sarajevo was indeed prosecuted by the Serbians, and as the Bosnian militias didn't have the same military resources, that was one reason why there were so many snipers causing trouble for everyone (military and civilian).

Most of this information was gathered in a course I took in 2008 so I could be fuzzy on some of the details. When I get home I can look up some legit sources. I also recognize that I am much more familiar with the reasons for the beginning of the conflict rather its end. It is a very complicated affair.

One book that I would strongly, strongly recommend on the subject of the Balkan Wars is Joe Sacco's masterpiece of comics journalism, "Safe Area Goražde."

Edit: TL;DR the serbs and the bosniaks were the two main "sides" with the Croats sort of staying off to the side, because they were less under attack by the Serbs. The UN Peacekeepers got involved but it was hard to come up with a solution because everyone was claiming everyone else's land, and then militias were trying to engage in ethnic cleansing to make reality reflect their claims.