Text messaging or texting refers to the exchange of brief written messages between fixed-line phone or mobile phone and fixed or portable devices over a network. While the original term (see below) was derived from referring to messages sent using the Short Message Service (SMS) originated from Radio Telegraphy, it has since been extended to include messages containing image, video, and sound content (known as MMS messages). The sender of a text message is known as a texter, while the service itself has different colloquialisms depending on the region: it may simply be referred to as a text or a texto in North America and Australia, text in the United Kingdom, an SMS in most of Europe, and a TMS or SMS in the Middle East and Asia.
Text messages can be used to interact with automated systems, such as ordering products and services for mobile phones, or participating in contests. Advertisers and service providers use texts to notify mobile phone users about promotions, payment due dates and other notifications that can usually be sent by post, e-mail or voicemail.

In straight and concise definition, "Text Messaging" by phones or mobile phones should include all 26 letters of the alphabet and 10 numerals, i.e. alpha-numeric messages or text to be sent by texter or received by textee

http://en.wikipedia.org/wiki/Text_messaging