Technically, the Fed funds rate is set by the banks themselves, not the Federal Reserve. However, the Fed can use its open market operations to put pressure on the banks to raise and lower the rate. The Fed simply purchases assets from the banks (usually Treasury notes) and places a credit on the bank's reserve account. Now the bank has reserves it doesn't need to meet its requirement. Therefore, it's willing to lower its Fed funds rate to lend the extra reserves to other banks. For the most part, the Fed funds rate rarely varies from the target rate.
However, the Fed has other tools it uses to reinforce the Fed funds rate. It can change the discount rate, which is what it charges banks who borrow directly from its discount window. These are some of the tools the Fed used to restore liquidity in the 2007 Banking Crisis.
The Fed funds rate is used by banks to set their other short-term interest rates. They charge a bit more for LIBOR, which is what they charge each other for one-month, three-month, six-month and one-year loans. It's also a base for the prime rate they charge their best customers. Banks use the Fed funds rate as a base to determine the interest rates they pay on NOW checking accounts, CDs and savings accounts. It also guides the higher interest rates they charge for bank loans, credit cards, and adjustable-rate mortgages.