What is the difference between the public sector and the private sector? what is a market economy regulated by? what is a planned economy regulated by?

The public sector is the part of the economy that is funded and administered by the government. The private sector is the part of the economy that is controlled by free and private interests. 

A market economy is regulated by the private sector, more specifically by market forces of supply and demand. A planned economy, in its purest form, is completely controlled by the government, meaning the public sector makes up the entire economy.

 In reality, there is no country whose economy is purely market-controlled or planned.