The term i18n is a shorter version of the word internationalization, an i +18 characters+ an n. L10n is a shorter version of localization. The L is capitalized so it is not confused with the i in i18n. These are called numeronyms or number based words. Like "Give me the 411" which is how I, in uptight jive, might say, "Tell me about it."
Internationalization refers to the work of enabling software to be language neutral. Localization is the work required to take that language neutral software and customize it for a particular language/culture. This includes language strings, date and time treatment, reading direction, and any data required to convey thoughts. So first you internationalize software, then you localize it into the specific languages you are targeting. The terms internationalization and localization came into use in the mid 1980's and floated around for awhile with various terms coming in and out of fashion until the industry settled on these two in the 1990's. I remember the term globalization being bantered about in the mid 1990's here at IBM, but, according to the Globalization101.org this now refers to economic or political integration rather than software integration.
The origin of i18n is a little more interesting. According to I18nGuy:
"A DEC employee named Jan Scherpenhuizen was given an email account of S12n by a system administrator, since his name was too long to be an account name. This approach to abbreviating long names was intended to be humorous and became generalized at DEC. The convention was applied to "internationalization" at DEC which was using the numeronym by 1985.
The extension of this naming convention to the terms Localization (l10n), Europeanization (e13n), Japanization (j10n), Globalization (g11n), seemed to come somewhat after the invention of "i18n". The terms Canonicalization and Normalization, defined more recently, also have numeronym forms (c14n and n11n)."
Ah yes, computer room humor. Wisely, very few of us are giving up our IT days jobs to travel the stand-up comedy circuit.