An exchange is a marketplace where financial instruments are traded. They can be either physical, like the New York Stock Exchange, or purely digital.
By centralising the buying and selling of a particular instrument, exchanges aim to make trading fair for all and as efficient as possible. The rise of electronic trading has led to huge progress in efficiency, and also changed the way that exchanges work in a major way.
Stock exchanges will have requirements for any businesses seeking to get listed. Most traders will not trade directly with exchanges: instead placing trades via a broker.