system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: why did western nations bail out their banks? What were the consequences if they had not?
gpt
The bailout of 2008 was premised on the notion of Keynesian economic theory, which basically means that economic cycles can last longer than human lifetimes hence government intervention into capitalist systems are necessary in the "short term". 

At first, the administration toyed with the idea of allowing the banks to fail, then letting the system repair itself. That's why Bear Stearns and Lehman Brothers weren't bailed out at first. It's important to recognize that shareholders were screwed anyway; in a bailout you basically dilute all your equity. So a bailout was less of a handout than a genuine rescue of the economic system. 

However, the intertwining relationships between banks and major financial "too big to fail" entities like Fannie Mae or AIG meant that allowing one bank to fail would mean a cascading domino effect across the board. The way it works is like this. Many banks work essentially by taking short term debt (deposits, money instruments, etc) and investing it into longer term debt (mortgages, bonds, etc). Loss of confidence would spark a liquidity crisis much like a global bank run on US financial entities; and with many banks reliant on each other to pay back their loans, one big default could mean everyone would fall like dominos. This was because their assets were marked to market; in layman's terms, they were worth whatever the bond market thought they were worth, so overnight they could become "worthless" and therefore bankrupt, even if nothing had changed in one day. (fundamentally, they were supposed to be bankrupt in an orderly fashion a long time ago)

Hence the immediate bailout response (TARP) was more of a liquidity measure rather than a fundamental rescue (there was nothing to rescue anyway) to give confidence to the international pension fund and institutional money managers, so they didn't do a "bank run" on the US economy. This allowed for a slower, more controlled release of the pent up financial panic, and gave some breathing room for the government to come up with a solution.

The next order of bailouts that the general public is more familiar with, Quantitative Easing, is basically the same thing but on a much grander scale. Essentially, nations themselves can become over leveraged with debt, although in a slightly different meaning as they also can print money. But the basic idea was that the US economy as a whole had far more credit in the system than was represented by the productivity of its population - which means some lenders had been suckered and weren't getting their money back. 

In order to "deleverage" the economy as a whole, you had to remove the debt in the system. This was done by buying up toxic assets from aforementioned financial entities and retiring them, using printed money since the government unlike most financial entities can print money. Governments have a host of options available to them that goes beyond the scope of this topic, but generally speaking the idea is the same. Remove debt so that the amount of credit in the system more closely matches actual productivity.

As there is just way too much debt accumulated over decades (started since around the 80's), there is no easy solution. Going against the Keynesian model and doing things the traditional way by letting the economy fail and correct itself had actually been done before - following the Great Depression of 1929. This had disastrous results for everyone, sparking mass suicides and social unrest, and the economy languished for over a decade and would have continued to do so for who knows how long had it not been for massive war spending. As Keynes himself put it, "in the long term, we're all dead anyway." This formed the basis for the bailouts, justified by the premise that it was the lesser of two evils.

To the common man on the street, the main thing you should be taking away from this is that the bailouts were necessary, but things aren't going to get much better soon. Adjust your expectations lower, as this will be the new normal going forward, at least for 10 years or so.