A mail server (sometimes also referred to an e-mail server) is a server that handles and delivers e-mail over a network, usually over the Internet. A mail server can receive e-mails from client computers and deliver them to other mail servers.
A mail server can also deliver e-mails to client computers.
Simple Mail Transfer Protocol (SMTP) is an Internet standard for electronic mail (email) transmission. First defined by RFC 821 in 1982, it was last updated in 2008 with Extended SMTP additions by RFC 5321, which is the protocol in widespread use today.
A mail server is the computerized equivalent of your friendly neighborhood mailman. Every email that is sent passes through a series of mail servers along its way to its intended recipient.
Although it may seem like a message is sent instantly - zipping from one PC to another in the blink of an eye - the reality is that a complex series of transfers takes place. Without this series of mail servers, you would only be able to send emails to people whose email address domains matched your own - i.e., you could only send messages from one example.com account to another example.com account.