The bank that is responsible for monetary policies among other things that is needed to create conditions allowing a stable economy.
Central Banks of different countries do serve different roles and mandates, but nearly all share the common goal of monetary policy that targets certain growth rate, inflation, employment level and and sometimes exchange rate. They all have the common instrument of setting interest rates, which has can have significant impact on all these factors.
An example is the United State’s central bank, the Federal Reserve Bank (Fed), which is run by the Federal Open Market Committee (FOMC) that meets to establish monetary policies. It’s mandate as of 2010 is price stability and full employment, but nothing about the exchange rate. Some central banks such as the Fed and the Bank of Japan (BoJ) are completely independent of the government, while some such as the Bank of England (BoE) must report to the government.