Fractional reserve banking is not directly part of the crisis narrative but it is a constituent feature of system. It is important because it's breakdown mode is the dreaded "bank run". This is how monetary levels were supposed to be controlled.
What it is
Fractional reserve banking is the habit of NOT keeping all deposits in kind but just a fraction of them. You invest the rest make a profit and return it when you are done. If the good is fungible then you can in fact return it at any time since you can put equivalent things in place of the original, even if the original is working somewhere else. Romans would sometimes treat this then emerging banking practice as criminal, today it is enshrined in our laws and institutions. The FED maintains the system, prevents bank runs etc.
Say you deposit a art collection for safe keeping. The bank cannot engage in "fractional banking" and has to keep your good whole. But with fungible goods, such as oil, grains, gold or indeed cash money, and many clients, your average banker can lend out a big portion of it, keep what is needed to cover average daily demands (a fraction of it).
From a legal standpoint, it was debated in the past whether a fungible good deposited and whose redemption was "at any time" was the sole property of the legal owner. Or could a bank invest that capital instead of letting it sit idle in a corner. Today the matter has been legislated. A bank, by virtue of the federal reserve banking act, can in fact invest a large proportion of the deposits entrusted to it (up to 90%).
This is the mechanism that the FED uses to control the money levels, at least in principle. They do so by controlling the monetary reserves of the banks and imposing ratios of deposits to reserves or investments to reserves.
When people talk about the "money multiplier" model this is also what they refer to but seen from another angle. If you give 100 to a bank and they lend 90 then you have 190 monetary units in circulation. From 90 you lend the corresponding fraction etc. If you do so recursively, you find that you have created 10x the money that was deposited. This is the chief way the FED controls the monetary levels. Fractional reserve banking is an inflationary practice. See Securitization for how the banks innovated their way out of the limits imposed by our forefathers and monetary control by the FED may have been in fact limited, the main culprit being that M2 proceeds M0 these days (debt creation happens first, FED adjusts M0).
How much money? A: Not too little and not too much
So the ratios of deposits to reserves are legislated. At any time banks should have around 1/10th of the deposit money in reserves. Those reserves are monitored by the FED. By legislation we limit what the balance sheets of those banks should support. Not too much or you buckle, not to little or you are under-investing in your economy. If the ratio of 1/10, put in place during the depression proved too limiting for the go-go years, it turns out that 1/40 is a sure fire way to kill equity capital as a 2.5% variation on assets will void the most junior member of the capital table: equity capital. So the number must be legislated somewhere in between.
The new legislation must account for Securitization as after all it was the total amount of debt generated and distributed by the system that was important, not just what the banks held onto.
Greenspan argues in his defense that the FED had effectively lost control of the monetary levels. It is an admission that the combination of M0 levels and M0/M2 ratios were insufficient. The debt dog was wagging the fiat money tail after all.
PS: to test your understanding of the above, read this poem, by Macro-Man one of my favorite hedgy reads