Central banks aren’t quite the same as other banks. Their main job is to manage the stability of the financial system. This means they’ve got to keep price levels from increasing or decreasing too quickly. Central banks are also usually concerned with the overall state of the economy and want to keep unemployment down and growth up. The two main ways they do it are managing private banks and controlling how much money is in the economy.
First, we can think of the central bank as a bank for other banks. Just like you might have an account at a commercial bank, your bank has an account at your country's central bank. Just like you deposit and withdraw money at your bank, your bank deposits and withdraws money with the central bank. Central banks are also responsible for enforcing all kinds of banking rules and regulations. For instance, the central bank can force other banks to keep a certain level of cash on hand to keep them from running into trouble. During emergencies, the central bank is also expected to give struggling commercial banks bailouts so they don’t go bankrupt. All of this gives the central bank quite a bit of power, and makes it a lot like a super-cop for policing banks.
On top of this, central banks are also expected to manage the money supply. Generally, central banks think that increasing the money supply lets them stimulate a sluggish economy, while decreasing the money supply will slow down an overheating economy experiencing inflation. The famous comparison is that a good central banker brings spiked punch to a dull party, and then takes the punch away just before things get out of hand. The way central banks actually control the money supply is called monetary policy.