A cryptocurrency exchange is a place where you can exchange fiat currencies (CAD, USD, EUR, etc) for cryptocurrencies (Bitcoin, Ethereum, Litecoin, etc.) and vice versa.
An exchange also provides its users with a on-platform digital wallet to hold cryptocurrencies they purchase. The exchange is responsible for the security of the wallets they provide to their users. If a user decides to withdraw their cryptocurrency off the exchange and into their own personal wallet, then the security of the cryptocurrency is now the responsibility of the user.