When the Latin empire was conquered by the Italians.

Although Latin was spoken after the fall of the Roman Empipre, various Romance languages began to be widely used around the 9th century (although they were developed much earlier), essentially making Latin extinct as a spoken language. However, the Humanist movement in the 14th to 16th centuries finished off the job by making Latin more academic by trying to revive it to its old form (thus, making Latin extinct as a written language as well).

The Latin language was the official language of the ROman Empire. The city of Rome was sacked by the Vandals in 410AD. After that, while Latin continued to be used as a church language, and regained its status as the language of official communication when the various European states accepted Christianity, it ceased to be an everyday language. This was the time of the codification of most European languages and their rise to become official languages of the new European states.

Uh ... never. It's still the language of the Roman Catholic Church - and not just their liturgy: all their official documents, communications, and scholarly works are written in it. Latin is also still spoken by many antiquarian academics throughout Europe (though more and more they speak it for the same reasons some people speak Klingon, rather than for any practical purpose, but they do speak it). Though the last Roman Emperor in the West was deposed in AD 476, Latin continued to be the language of Arts & Letters, of diplomacy, of international business, of science, medicine, academia, theology, and law right up into the 1700s, when French supplanted for diplomacy, English for commerce, and (in the 1800s) German for science and technology. (When Newton wrote is proofs of the Law of Gravitation and the mathmatics of Calculus in the late 1600s, he wrote them in Latin.) Latin continued to dominate Law until Napoleon supplanted Justinian's Codex of Roman Law with the Napoleonic Code (in French, of course), but in Britain & areas under Common Law, Latin only slowly gave way to English (1850-1950). In an English courtroom even today, it isn't unheard of for the Barristers and Judge to break off in a discussion in Latin.

It's still alive in law, you know!! And to a lesser extent, philosophy.

You've already gotten sufficient answers about the Fall of the Roman Empire and the consequent emergence of other strong powers. The Roman Empire constituted a large part of Europe. When this dissolved, the opportunity for French, Spanish, Italian, Portuguese, and Romanian languages infiltrated what had been dominated by Roman culture. Latin never became extinct, though. It is still a very valuable curriculum for anybody entering medicine or law. Latin improves SAT scores by building English vocabulary - the amount of derivatives from Latin is just amazing.

Well, it didn't is the simple answer. It is very much alive here for example http://schola.ning.com Languages like Sumerian and Ugaritic are indeed dead languages. Latin is not, as there are several households across the world that are Latin speaking, and where the children grow up learning Latin. As there are many living languages with only a handful of speakers, Latin, with thousands of speakers across the world, and many more learning how to speak the language ( for example using this course : http://latinum.mypodcast.om ) it simply does not qualify as a dead language.