When: 800 BCE - the Renaissance. The so-called "golden" and "silver" ages of classical Latin spanned 75 BCE to the third century, and eventually the language transitioned into late and medieval Latin in later centuries.
Who wrote in it: Cicero, Julius Caesar, Cato, Catullus, Vergil, Ovid, Marcus Aurelius, Seneca, Augustine, Thomas Aquinas.
Pros to learning it: The Ryan Gosling of Dead Languages: it's popular and pretty sexy.
Cons to learning it: No one really speaks it anymore. (Except in a few diehard "Living Latin" communities, and at the Vatican, obvi.)