In a private flight, the pilot is not paid, and the aircraft owner/operator does not receive money for the flight (other than rent from the pilot, in some cases). In many countries, private aviation operates to less strict standards than commercial aviation. For example, in Canada and the United States, aircraft owners are allowed to perform basic maintenance tasks (such as oil or tire changes) on their own privately-registered aircraft, but only licensed mechanics may perform those tasks on commercially-registered aircraft. Private pilots normally are not required to demonstrate the same level of proficiency on their flight tests, and take fewer and less rigorous medical examinations.
It is the purpose of the flight, not the aircraft or pilot, that determines whether the flight is private. For example, if a commercially-licensed pilot flies a commercially-registered plane to visit a friend or attend a business meeting, most countries would consider that to be a private flight.