There are various candidates for the first email system.
The best is the mail system developed for MIT's Compatible Time-Sharing System. For more information, see a short article by Tom Van Vleck. On this basis, the first email was sent in 1965.
It is not known who the first person was to send an email message.
Wonderfully, Mr. V. A. Shiva Ayyadurai is said to have invented email in 1978, aged 14. Mr. Ayyadurai is clearly a very capable man. However, that assertion is incorrect. Google him for fun, or see this nice little article by Richi Jennings. --David Ferris, with thanks to Nathaniel Borenstein*
* David sent his first email in 1974, shortly after he used a full screen word processor. He immediately became an enthusiastic user of  both and thought they might, possibly, conceivably, have potential over the long term. Nathaniel was far more assertive and productive. He was partly responsible for defining the MIME email format, a key step in allowing the email users of the world to connect. And his assistance and advice were invaluable during the establishment of this museum.