An exchange is a business that facilitates trading between a buyer and seller, with no contact needed between them. The exchange service handles accepting deposits of funds and receives people’s buy/sell orders, which are then placed on the exchange’s order book. When the order price of a buy order (a.k.a. bid) on the order book matches the order price of a sell order (a.k.a ask) on the order book, the exchange automatically trades the funds and credits the buyer’s and seller’s accounts. The exchange will often charge both the buyer and seller a percentage fee per trade (which is how they make money). After the trade has been executed, the buyer and seller can withdraw their newly acquired funds.
Typically this service is offered as an online website. See the list of “Bitcoin Exchanges and Outlets”.