The Federal Reserve does not directly set most of the interest rates you encounter — your mortgage rate, credit card APR, or savings rate. What it directly controls is the federal funds rate: the target range for the overnight rate at which commercial banks lend to each other. By adjusting this target, the Fed indirectly influences virtually every other interest rate in the economy.
The mechanism has several components. First, the Fed sets its target range for the federal funds rate at FOMC meetings. The primary tool for keeping the actual rate within this target range is the interest rate on reserve balances (IORB) — the rate the Fed pays to commercial banks on the reserves they hold at the Fed. Banks have no incentive to lend to each other at a rate below IORB, because they can earn that rate risk-free from the Fed. This puts a floor under the federal funds rate.
The overnight reverse repurchase agreement facility (ON RRP) provides a secondary floor, allowing money market funds and other financial institutions to park cash with the Fed at a specified rate overnight. This broadens the floor beyond just commercial banks.
Open market operations — the buying and selling of government securities by the Fed's trading desk in New York — were historically the primary mechanism for adjusting the supply of reserves in the banking system, influencing the federal funds rate. In the era of large reserves since 2008, this mechanism has become less important than the IORB rate.
Market rates beyond overnight adjust in response to Fed policy through expectations. If markets expect the Fed to raise rates multiple times over the next year, two-year Treasury yields rise immediately to reflect those expected future rates. Long-term rates are determined by expectations of the entire future path of short-term rates plus a "term premium" for uncertainty.
This is why Fed communications — speeches, meeting minutes, the Summary of Economic Projections (the "dot plot") — move markets even when no rate change is made. Markets constantly update their expectations about the future path of rates, and these expectations are embedded in current long-term rates.