There was a great deal more economic freedom during that time than there is today, to be sure. However, even 'mostly' laissez-faire markets are still not true laissez-faire markets. Government intervention and monetary expansion will always cause an eventual depression, even if those interventions are small by today's standards. Another thing to note is that even though government distorts the market much more now than in earlier periods, you could say that they have gotten much better at building houses of cards.
During the 19th century, banks and the government were limited in their monetary expansion by being backed with gold and silver. However, during that period (1865–1912), banks were not limited to holding 100% reserves. Banks only had to maintain 25% reserves (for every $1 loaned, the bank had to have $.25 worth of gold/silver). This allowed monetary expansion, which caused a boom (similar to today's housing boom & bust) especially in the railroad industry. Eventually, every monetary expansion has to end, and in 1893, that period's expansion did. Before the Federal Reserve banking system, each individual bank had it's own level of reserves (with a minimum of 25%). There was a national currency at the time, but that didn't help. If a bank fails, it takes all of the fiat money it created with it. Imagine you had a $100 bank note. You know that if your bank goes bankrupt, you will be left with nothing but a worthless piece of paper. So, upon hearing news of a bank run, you would head to your bank as quickly as possible to redeem this note in order to not be left with nothing. Because your bank doesn't have enough specie (gold/silver) to pay its depositors with, it fails. Whoever didn't get to the bank in time was left with nothing. If, however, you had a note that was not issued by your bank (a national currency note), you could take this to any participating bank (or the treasury) and get your gold. However, once one bank has failed, you would likely become worried of continued bank failures, which would leave you with a worthless note if they all failed (or suspended payments, as they often did). So, as both you and others demand payment in specie for the paper notes you hold, more banks fail (because they only keep 25% reserves). This all leads to a domino effect, which can take out all but the most financially responsible banks. Of course, if the banks had 100% reserves as they should have, there would never have been a problem.
In a fractional reserve system, when a bank goes bust, all of the fiat money that was created by that bank is no longer worth anything, causing monetary contraction. The more banks fail, the more monetary contraction, which of course leads to more bank failures, and the cycle continues. Once started, monetary contraction makes loans more expensive (due to money being more scare), which causes business activities that were profitable at a lower interest rate to be no longer profitable.
For example, let's say I'm building a railroad. The bank offers me a loan at 5%, and I figure that I can build this line, pay for maintenance and labor, etc, and end up with, say, that 5% I will need to pay and 2% profit. Well, if banks fail, and loan costs rise to say 8%, I would no longer be able to make a profit by building this particular line. In order to minimize my losses, I would need to reallocate all of the resources that were put in to this project, which would of course probably lead to lay-offs, etc. As painful as that sounds, those resources need to be put to a better use that will generate a profit, in order to be sustainable and provide long term economic growth.
So, the cycle goes: first the monetary contraction, leading to bank failures, leading to unemployment and a recession.
This is a very basic over-simplification of what happens, but I'm not an economist so it'll have to do.
Here's a quick rundown of Austrian Business Cycle Theory, which explains what happens in more detail:
http://mises.org/daily/672
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