The Federal Reserve was unprepared for the depth of the disaster of the stock market crash late in 1929. It was a decentralized institution, split into districts with different governors. Although these governors understood that they should coordinate their efforts and tried to do so, they could not agree about the best solution to the crisis.
Since so much money was wiped out by the stock market crash, the supply of money in the US economy fell by almost 30 percent between 1930 and the winter of 1933. Because people had so much less money to buy goods, the price of goods naturally dropped by an equal amount, a significant deflation. This was destabilizing for the economy. For example, if you figure prices will go down, then you will wait to make purchases. If you have debt, then you are going to run into trouble. For instance, if the value of your house dropped from $10,000 to $7,000 and you owed $8,000, you could not very well sell the house to clear the debt. Also, with less money available (this was before the days of widespread consumer credit) people had less to spend, which contributed to a severe downward spiral: as businesses sold fewer goods, they had less money and so they laid off more people.
The Federal Reserve's biggest failure was its inability to prevent the deflation. It could have, for example, lent banks money so they did not collapse and wipe out people's savings or they could have printed more money. Normally printing money causes inflation, but in this case it would have merely prevented deflation.
As stated in the first paragraph, people with the power to change the course of the economy were surprised by the collapse and were confused about what had gone wrong. Decentralization meant nobody had sufficient authority alone to do what needed to be done in order to prevent deflation. Some thought raising interest rates and reducing the amount of money in the economy, although it was deflationary, was the cure. Now we know that was wrong, but hindsight is 20/20 and the people in charge had to deal with an unprecedented crisis in real time without a reliable road map.