First things first, there's a few answers to this on previous posts that I'm sure someone will point to as it's an oft-answered question. However, being as it's the first time I've got in a thread early enough for that not to have been put as a comment already, I'll put my own full answer as follows...
There is no one reason why Yugoslavia fell but we can broadly cluster the reasons into three key areas - the structure of the state itself, the economic situation of the 1980s and nationalism. All of these link to each other in a way but for the sake of ease, splitting things into those areas will make this all a bit easier to understand without going into massive depth! To go through bit by bit...
Economic issues
For a start, come the end of the 1970s, Yugoslavia had amassed a lot of debt because of how the economy itself was built. It was reliant on external investment based around Yugoslavia's unique balance between East and West meaning that they could get investment from a wide range of sources. But, after WW2, Yugoslavia was also wrecked both in terms of the rigours of war itself but also because the Kingdom era saw a lot of things centralised around Belgrade. So all of this investment was needed to diversify the locations of the economy with much of the investment going towards the republics that had been neglected in the Kingdom (namely Croatia and Slovenia). The result of this was that, throughout the long boom, Yugoslavia's economy grew massively with two key weaknesses - inflation and unemployment. Unemployment was there but was kept in check because there was a steady outflow of workers going to Germany and Sweden, but was present because the state of education pre-1945 was such that Yugoslavia had a massive skills gap.
So, come the Oil crisis of the Mid-70s, investment from outside dried up and Yugoslavia needed to take on debt to keep the wheels turning - between 1973 and 83, debt doubled with inflation going from 20% (already pretty high!) to 50% with the trade deficit increasing even more rapidly. Come the death of Tito, these foreign debts start to be called in.
But, above I mentioned that the economy was regionalised post-WW2. What that did was swap the Kingdom era complaints of everything being centralised in Belgrade and things being unfair to Croats, etc around 180 degrees. Yugoslavia quickly developed a north-south split with GDP in the north being as high as four times that in the likes of Skopje and Prishtina. This, of course, wasn't sustainable as is seen quickly in the 1980s with the protests in Kosovo.
Finally, the USSR. If Yugoslavian success was based around playing East off vs West then, when the 80s hit and the Soviet economy starts running into trouble and becoming less of a threat, the stimulus for Western involvement (ie to keep Yugoslavia Western facing) suddenly dried up with it.
As such, you have a nation that is already in some economic trouble because of how the economy was built suddenly having all of the factors collide and exacerbate at once - external investment failing to come back after the Oil crisis and Yugoslavia having accumulated a large debt trying to run the economy without it. There was also plenty of corruption, such as the AgroKomerc scandal, that showed to a lot of people that the system just wasn't working.
Structural issues
Going back to the early 70s and the Croatian Spring, a new constitution was drawn up. This solved some things at the time but left some glaring flaws.
The main one as things happened was that Kosovo and Vojvodina had their voting rights in the presidency increased so the vote came out as eight republics rather than six. This, ostensibly, was because Serbia was too big a republic and ran the risk of being outvoted so much as to sow dissent there. Quite simply, it balanced north and south. The problem was, as we hit the late 80s, was that were an unscrupulous politician to come along who wanted to consolidate power around themselves in Vojvodina and Kosovo (which had large Serb populations), then Serbia would become three in eight rather than one in six and, if their natural allies of Montenegro joined in, then Serbia's vote suddenly became a veto as they'd control four of eight at most rather than two of six at most. This would be exactly what Slobodan Milosevic did.
In addition, 1974 set up the succession plan for Tito's passing which was to go to one year rotating system. Yugoslavia went from one omnipresent man to a succession of people not around long enough to ever really be memorable or do much of anything.
Finally, of course, there was Tito himself and, while Great Man theories usually aren't... well, great, Tito lays more of a claim to their legitimacy than most given what happened after he passed. Without spending too long on it, both the Kingdom era and the post-Tito era lacked a dominant figure such as Tito and look what happened to them. Tito may not have held the nation together through force, such as Hoxha did, but there was very little prospect of serious internal opposition while he was around given that Tito was, essentially, above such disputes as nationalism.
Nationalist issues
Here comes the fun part. While nationalism isn't necessarily a popular theory as to why Yugoslavia fell apart, it does also neatly cover why Yugoslavia was always going to be prone to falling apart should other things go against them.
While I wouldn't presume to make your mind up for you, I'd note my opinion very much lands on the concept that Yugoslavia was always an absolutely terrible idea. If one goes back a century to Illyrian or Croat or Serb nationalism, one would note that none of these ideas featured one massive pan-Slavic state. The Kingdom of Yugoslavia was formed out of WWI as a compromise between a Serbia that was broken so badly by war it couldn't possibly stand as it's own independent nation and a collective between Slovenia and Croatia in the North that wanted to break from the broken Austrian empire. With a Serb nationalist King for much of the time, that Yugoslavia was marked by pushes from the centre for it to become Velika Srbija (Greater Serbia) and resistance against that from the Northern and Muslim communities. Tito's Yugoslavia went the other way, as mentioned with the economy, and played into the hands of some who felt that the state policy was that a strong Yugoslavia could only be achieved with a weakened Serbia. Hence my contention would be that Yugoslavia was just a bad idea.
Beyond that, in the post-war era, old beefs simply weren't squashed. After all, Croatia did commit plenty of genocidal acts against Serbs during WW2 but, afterwards, there was an expectation that everyone just sort of got along, especially as many of those most responsible for the genocide either a) fled (with many being hunted down by intelligence agencies) or b) met an end in reprisals such as Bleiburg and Bloody Christmas. 1974 placated Croatian nationalists constitutionally while also repressing them in other aspects of culture. Kosovo was also an issue which exacerbated during the 1980s.
All of these descriptions are fairly brief summations of the main drivers behind the decline of Yugoslavia to give you an idea of what factors were around in, say, 1985 as Yugoslavia was about to accelerate from a weak state to a failed one, because that went very quickly and is another post for another time. If there's any part of that, you'd like expanding on, I can do so in the comments (when I've a little more time as to go through all fully would be a mammoth comment!) or if there's any disagreements.
What I'd note is that there is no one reason and every commenter will have a different idea on this. I would say that the nation itself was always so weak thanks to the ethnic divides and its rather clumsy path to existence that failure was always close and that it was only a combination of the force of will of Tito and the good fortune he had in passing away before the mistakes of the 1970s really hit home that kept Yugoslavia going for so long and kept his reputation fairly intact.