This is a very common question and is covered in the FAQ.

To add to what's there: Latin didn't die. No one ever stopped speaking Latin. Instead, different groups in different places continued speaking what they had been speaking, and these different varieties, which were already diverging well before the end of the Roman Empire, continued evolving. Eventually they evolved into the modern Romance languages, many of which are not intelligible with one another.

We can contrast this with modern languages undergoing language death where it's really clear that over the span of even just two or three generations, a community can shift its entire language. Okinawan, a language I do work on, had its last monolingual speakers four or five generations ago, and the current generation (people under 40, let's say) has basically no command of the language at all, instead being monolingual Japanese speakers.

Medieval Latinist here. The top post is correct that Latin never really died because it morphed into the Romance languages.

But if one wants to identify a specific time when Latin began to be studied in a "dead" form, it would be during the Renaissance. The Middle Ages got by with a vulgar form of spoken Latin that was the cross-cultural language of diplomacy and the Church. During the Renaissance, however, humanist scholars rediscovered many of the great Latin classics - Cicero's letters, Seneca's tragedies -, idealized its rhetoric and taught it in the schools as the only "proper" form of Latin. Ciceronian Latin was impossible to speak (no one ever actually spoke like Cicero wrote) and so it became a dead scholarly language while the medieval vulgar Latin died away and was replaced in international circles by French (i.e. lingua franca).