This quasi-public institution acts as central bank for the United States of America.
This means that it regulates the banks, by setting reserve requirements, how much money the bank must have on hand to pay depositors who want their money. Raising the requirement reduces the amount of money in circulation and lowering it increases the amount in circulation, but this is rarely done since it has a very strong effect upon the money supply due to the money multiplier effect.
It sets the discount rate, the interest rate charged by a Federal Reserve Bank on short-term loans to depository institutions. The discount rate is set every two weeks by the directors of each Reserve Bank, subject to the approval of the Board of Governors.
To effect the market in more subtle ways it conducts open market operations. They are the purchase or sale of U.S. Government securities in the "open market" (also known as the secondary market) to depository institutions, like banks. This increases or reduces the amount of reserves the banks have to loan out and thus inversely effects the money supply.
It consists of the 7 member Board of Governors, which sets over all bank policy and orders open market actions with advice from the Federal Open Market Committee. Each member is elected to one 14 year term and no more than one member can come from any of the 12 Federal Reserve Districts.
Under them are the twelve Federal Reserve Regional Banks. Each of which is a private institution owned by member banks from within that district. Under them are the 25 Federal Reserve District Banks based in many of the larger cities in the US.
All national banks must be part of the Federal Reserve System and buy stock in the Federal Reserve. State Banks can elect to be members or not, as they choose, though state banks have to meet stricter requirements and thus are generally stable even if they elect not to join.
As with most powerful organization the Federal Reserve is hated and feared by many. Due to its private nature most of the accusations leveled at the Fed are that it are that it is robbing the American public. The two remedies most often proposed for this are returning to using a gold standard and nationalizing the Federal Reserve Bank. (Ignoring the various problems with such systems, of course.)