SMS messaging--where text messages are sent on mobile phones--was first used in December 1992. Neil Papworth, a 22-year-old test engineer for Sema Group, used a personal computer to send the text message “Merry Christmas†via the Vodafone network to the phone of Richard Jarvis. (source: Wikipedia).
Curator's note: we say "text message" here in its received sense, that of cellphone-based text. Clearly, computer-based textual messages were sent much earlier, in the mid-1960's.--David Ferris, with thanks to Wikipedia