There are still many people who persist in propagating the myth that the Great Depression represented a failure of the capitalist system that could only be solved by active government intervention in the economy. Like all myths, the empirical data does not support that belief even as the belief has continued to reside in the superficial understandings of many people.
Lawrence Reed at the Mackinac Center for Public Policy wrote Great Myths of the Great Depression, a very approachable document for reading by the layman.
Milton Friedman won his Nobel Prize in Economics in part for his book, Monetary History of the United States, 1867-1960, in which he offered the first rigorous explanation of what caused the Great Depression.
In the third chapter of his book, Capitalism & Freedom, Friedman offers some highlights of what is in his more indepth study of the Great Depression:

However, in view of the importance which the Great Depression of 1929-1933 played in forming – or, I would say, deforming – general attitudes toward the role of government in economic affairs, it may be wroth indicating more fully…the kind of interpretation suggested by the evidence.
Because of its dramatic character, the stock market crash in October 1929, which terminated the bull market of 1928 and 1929 is often regarded as both the start and the major proximate cause of the Great Depression. Neither is correct. The peak of business was reached in mid-1929…
For something like the first year, the contraction showed none of those special features that were to dominate its later course. The economic decline was more severe than during the first year of most contractions, possible in response to the stock market crash plus the unusually tight monetary conditions that had been maintained since mid-1928…
…it is clear that the Reserve System should already have been behaving differently than it did, that it should not have allowed the money stock to decline by nearly 3 percent from August 1929 to October 1930…
The character of the contraction changed drastically in November 1930 when a series of bank failures led to widespread runs on banks, which is to say attempts by depositors to convert deposits into currency…
Prior to 1930, there had been no sign of a liquidity crisis, or any loss of confidence in banks. From this time on, the economy was plagued by recurrent liquidity crises…These [runs on banks] were important not only or even primarily because of the failures of the banks but because of their effect on the money stock…
This was precisely the kind of a situation that had led to a banking panic under the pre-Federal Reserve banking system…
…one of the major reasons for establishing the Federal Reserve System was to deal with such a situation…
The first need for these powers and hence the first test of their efficacy came in November and December 1930 as a result of the string of bank closings…The Reserve System failed the test miserably. It did little or nothing to provide the banking system with liquidity…It is worth noting that the System’s failure was a failure of will, not of power…the System had ample power to provide the banks with the cash their depositors were demanding. Had this been done, the bank closings would have been cut short and the monetary debacle averted…
The [economic] figures for the first four or five months of 1931…have all the earmarks of the bottom of a cycle and the beginning of revival.
The tentative revival was however short-lived. Renewed bank failures started another series of runs and again set in train a renewed decline in the stock of money. Again, the Reserve System stood idly by. In the face of an unprecedented liquidation of the commercial banking system, the books of the [Reserve System] show a decline in the amount of credit it made available to its member banks…
After more than two years of severe economic contraction, the System raised the discount rate…more sharply that it has within so brief a period in its whole history before or since….It was also accompanied by a spectacular increase in bank failures and runs on banks…
A temporary reversal of policy in 1932 involving the purchase of $1 billion of government bonds slowed down the rate of decline. Had this measure been taken in 1931, it would almost surely have been sufficient to prevent the debacle just described. By 1932, it was too late to be more than a palliative and, when the System relapsed into passivity, the temporary improvement was followed by a renewed collapse terminating in the Banking Holiday of 1933…A system established in large part to prevent a temporary suspension of convertibility of deposits into currency – a measure that had formerly prevented banks from failing – first let nearly a third of the banks of the country go out of existence and then welcomed a suspension of convertibility that was incomparably more sweeping and severe than any earlier suspension…
All told, from July 1929 to March 1933, the money stock in the United States fell by one-third…it is literally inconceivable that money income could have declined by over one-half and prices by over one-third in the course of four years if there had been no decline in the stock of money. I know of no severe depression in any country or any time that was not accompanied by a sharp decline in the stock of money and equally of no sharp decline in the stock of money that was not accompanied by a severe depression.
The Great Depression in the United States, far from being a sign of the inherent instability of the private enterprise system, is a testament to how much harm can be done by mistakes on the part of a few men when they wield vast power over the monetary system of a country…

ADDITIONAL INFORMATION:
In the comments section below, Marc has added some important links that further elaborate on the causes for the Great Depression. In addition to the Lawrence Reed piece from the Mackinac Institute referenced above, Marc guides us to:
The Government and the Great Depression by Chris Edwards, Director of Tax Policy at the Cato Institute
The Fed’s Depression and the Birth of the New Deal by Paul Craig Roberts and Lawrence Stratton
You can also learn more by reading FDR’s Folly: How Roosevelt and His New Deal Prolonged the Great Depression.
ADDITIONAL INFORMATION II:
Professor Don Boudreaux from Cafe Hayek has more.