There are banks and banks. Some banks only serve banks. And, above all of them, some banks give instructions to the rest of the banks. Those are the central banks.
A central bank is a state institution that is in charge of several essential aspects of the economy, not only of the country itself but, depending on the size of the economy it regulates, it can affect -and a lot- the rest of the surrounding countries. And, in addition, it has an important task: it is the body in charge of issuing the money, although it is not necessarily the one who prints it.
For example, until 1874 in Spain there were private banks that issued their own money. From that date and following the trend of other countries in the world, it was decided that only the Bank of Spain would be empowered to issue currency. Today only extremely weak states, such as Somalia, allow banks or private businesses to issue their own money.
This is important because the uncontrolled issuance of money can lead to a devaluation of the currency it represents and an inflation crisis, so central banks try to have as much control as possible over the amount of money in circulation. Before, the money was the equivalent of a quantity of the gold that the central banks had (and have) in their vaults. Money now represents something like the value of the economy it supports, so it's very important to protect that value and keep it stable.
But, then, what functions does a central bank have?
As a general rule, central banks are:
Custodians and managers of gold and foreign exchange reserves; that although they no longer serve to determine the value of their currency, they do serve for other types of financial operations.
Providers of legal tender money; As we have said before, they are the only ones in charge of printing or having money printed. And, for this very reason, they are also in charge of carrying out monetary policies in their area and of maintaining price stability as much as possible.
Executors of foreign exchange policies; that is, they are in charge of setting the exchange rate of their currency concerning others.
Providers of Treasury services and financial agents of the Public Debt of national governments; which means that they are the bank that the State uses for its financial operations and those who borrow in its name through bonds, letters, obligations, etc.
Government advisors, in the reports or studies that are appropriate. Although central banks are generally independent of governments, they have a voice and sometimes a vote in their economic policies.
Auditors are in charge of carrying out and publishing the statistics related to their functions; Due to their responsibility concerning monetary policies, central banks collect a huge amount of economic data that serves to know the state of finances in the area over which they have jurisdiction. They also inspect private banks to make sure they are solvent and that their citizens' savings are safe.
Bank of banks. When private banks need unusual loans, they can go to their central bank of reference, which can also offer it to guarantee that they have enough capital so that their clients do not lose their savings. In the same way, just as a private bank can seize our properties, the central bank can intervene in a private bank in case of bankruptcy or mismanagement.
Supervisors of the correct functioning and stability of the financial system, as well as payment systems and the banking market. Central banks ensure that the entire banking system functions according to the rules, actively inspecting everything from the way mortgages are granted to the operation of credit cards.