Download audio version here
Email is much older than ARPANet or the Internet. It was never invented; it evolved from very simple beginnings.
Early email was just a small advance on what we know these days as a file directory - it just put a message in another user's directory in a spot where they could see it when they logged in. Simple as that. Just like leaving a note on someone's desk.
Probably the first email system of this type was MAILBOX, used at Massachusetts Institute of Technology from 1965. Another early program to send messages on the same computer was called SNDMSG.