Due to its informal character, there was neither well-established standard nor common name. In the early days of e-mail, the humorous term "Volapuk encoding" (Russian: кодировка "воляпюк" or "волапюк", kodirovka volapyuk) was sometimes used.
More recently the term "translit" emerged to indiscriminately refer to both programs that transliterate Cyrillic (and other non-Latin alphabets) into Latin, as well as the result of such transliteration. The word is derived by convenience truncation of the term transliteration, and most probably its usage originated in several places. An example of early "translit" is the MS DOS program TRANSLIT by Jan Labanowski, which run from the command prompt to convert a Cyrillic file in a Latin one using a specified transliteration table.
There are two basic varieties of romanization of Russian: transliterations and Leetspeak-type of rendering of Russian text. The latter one is often heavily saturated with common English words, which are often much shorter than the corresponding Russian ones, and is sometimes referred to as Runglish or Russlish.
The main reason that transliteration is used with Russian is that in text messages you get more Latin characters for your money: usually 160 Latin characters per charged message versus 60/70 Cyrillic characters. Obviously the onus is on getting one Latin symbol (of which there are 26) for each Cyrillic symbol (of which there are 33 in Russian, and extra symbols in Ukrainian and other Cyrillic-based languages). Only those used for Russian are exemplified here.
(Where variants are given, the first is most common and the last is less common - although trends change quickly and differ from person-to-person.)