What service is used when your email program exchanges messages with the mail server on a distant network?

The answer is SMTP (Simple Mail Transfer Protocol)

SMTP is a TCP/IP protocol used in sending and receiving emails. This service is usually used with two other protocols, IMAP or POP3. Most email systems nowadays used SMTP to send messages from one server to another. Generally, an SMTP is used to send messages from a mail client to a mail server.