Although His Holiness Pope Francis might have a private or official e-mail address, he does not have a publicly listed email address. Those who wish to contact him by modern means are not relegated to snail mail, however; he does have an active  Twitter feed.
For contacting Pope Francis by traditional mail, the Vatican provides this address:
His Holiness, Pope Francis
Apostolic Palace
00120 Vatican City
(Don't add "Italy" to the address; the Vatican is a separate political entity from Italy.)