The first version of a computer email system that resembles anything like what we have today showed up around 1965. Computers had been primitively networked a few years earlier, and you could technically send and receive content.

At MIT in 1965, Tom Van Vleck and Noel Morris wrote the mail command for the Compatible Time-Sharing System (CTSS) which allowed users to send mail (a file) to each other. When the recipient logged in, they would be notified and could print the contents of the email.

In 1967, "messaging" was on the list of the reasons for developing ARPANET, the precursor to the Internet. The other reasons included: load sharing, data sharing, program sharing, and remote logins.

In 1969, Tom Van Vleck would reimplement the mail command for Multics. Neither version did anything about privacy, authentication, or security. Any user could write or view any other user's mailbox.

The U.S. Postal Service experimented with forms of electronic mail in the 1960s and 1970s. An effort called MAILGRAM between USPS and Western Union sent messages electronically to a post office where they were printed out and mailed as normal letters.

In 1971, the SNDMSG and READMAIL programs for TENEX allowed users to specify the traditional Subject:, To:, and cc: headers. It also introduced the @ sign to refer to an ARPANET host.

The same year, a mail command was added to Unix (another copy of Tom Van Vleck's implementation for Multics and CTSS). The Mail Box Protocol (RFC 221) was designed.

In 1974, MSG was the first mail application that included features like Reply and Forward. The Unix MBOX format was developed. Attachments came in 1976.

Later:

You can see the entire timeline until 2011 here.