How Bitcoin wallets and it's transfer process work?
First, a wallet is a software application that usually manages your keys, addresses, and transactions. When someone sends you Bitcoin, it is generally sent to one of the addresses that the wallet software manages. There really is no limit to how many wallet software applications you can run, and not really a limit on addresses either (except for disk space, but that's not really a concern etc.).

Your wallet software should allow you to create new addresses that you can use to receive Bitcoin.

Exchanges work by managing your Bitcoin in their internal wallet software, and you exchange anonymously with others by placing and filling orders on the global order book. The exchange manages the Fiat (dollars) currency exchange internally as sort of an escrow.