Latatian is a (presumably) dead language, originating in Sto Lat, from which a great deal of Morporkian is derived. It translates smoothly to Roundworld's Latin, as Morporkian does to English. Its primary uses in the Century of the Anchovy are for legal jargon from the Guild of Lawyers and to allow the College of Heralds to make excruciating puns. The Latatian alphabet is used in most of the languages Hubwards of the Circle Sea.
Remaining texts and inscriptions from Ankh-Morpork's early history tend to be in Latatian. This may be because Latatian remained the language of educated discourse while the masses spoke some version of Morporkian: a transition point can't be determined accurately. The languages of Quirm and Brindisi obviously derive from Latatian, suggesting that Latatian was spoken through some of the period of the Morporkian Empire.
Bear in mind that most of the Latatian quoted in the Discworld books has been made up comparatively recently by people who hadn't studied much, and many of the common expressions are not exactly as accurate, er, perfectly...well, they're wrong. Many use Morporkian words fitted with Latatian trim, as some on Roundworld try to speak Spanish by adding an "o" to the end of the English words. Dog Latin is probably heard more often than real Latin on Roundworld as well.
The language survives in ecclesiastical and legal terminology, and as on Roundworld, several months of the year are survivals of Latatian months named in honour of great leaders and emperors: Grune, August, etc.
In Pyramids, it is misspelled as "Laotation"- an "o" has been added in the beginning, and the last "a" has morphed to another "o". Since this is the only usage of "Laotation" instead of "Latatian", we can assume that Teppic actually speaks this language, not "Laotation". However, the official Discworld Mappe shows a nation called Laotan in the Mountains of the Sun, it is also possible that the above is a reference to a language of this region.
See Latatian Phrases for a list of expressions still used on Discworld.