In the public sector, the government owns most of the assets and provides all the services. The purpose of the public sector is not just to earn profits. Governments raise money through taxes and other ways to meet expenses on the services rendered by it. Railways or post office is an example of the public sector.
In the private sector, ownership of assets and delivery of services is in the hands of private individuals or companies. Activities in the private sector are guided by the motive to earn profits. To get such services we have to pay money to these individuals and companies. Companies like Tata Iron and Steel Company Limited (TISCO) or Reliance Industries Limited (RIL) are privately owned.