The main feature of the private sector is that it operates outside of the state, or government. It is owned, funded and run by private entities. This can be one person, often known as a Sole Trader, or a group of people, sometimes referred to as a Corporation.
The objective of the private sector is to make a profit. Where the public sector may be funded to provide a service, the private sector may provide this same service, but with the intention of making money.
A good example of this is healthcare in the UK.
The NHS is the public sector provider which receives money from the government- which of course is indirectly funded by us, the general public, through income taxation- and provides healthcare services for free. Whereas private healthcare providers like Bupa, for example, offer private healthcare, which may be the same services - but you have to pay for it (and this means you have much shorter waiting times for treatment, and other benefits).