Email servers manage sending and receiving messages through a system of protocols and components. When a user sends an email, it is transmitted via the Simple Mail Transfer Protocol (SMTP) to the sender's email server. The server processes the message and determines the recipient's server, delivering the email accordingly. For incoming messages, servers use Post Office Protocol (POP) or Internet Message Access Protocol (IMAP) to retrieve emails from the recipient's server. These protocols allow users to access and manage messages efficiently, ensuring proper delivery and organization of emails across various platforms.