Mobile phone texting, or SMS messaging as it is more formally known, was first used in December 1992. Neil Papworth, a Sema Group engineer used a personal computer to send the text message “Merry Christmas” via the Vodafone network to the phone of Richard Jarvis (source: Wikipedia).
Widespread European use began around 1999, with near-universal use around 2003. The main promoter of early SMS in Europe was the cost difference between SMS and voice.
Widespread US use began around 2004, with near-universal use around 2006. In the US, the cost of mobile voice was considerably lower, reducing the attraction of SMS. In addition, technological impediments existed: the CDMA and TDMA standards in use by Verizon and AT&T had to adapt SMS from the GSM standard already being rolled out in Europe. This delay cascaded into later network capability and handset capability.–David Ferris, with thanks to Wikipedia and David Gross