One of the stages in the Latin
language, and the variety in which almost all its great literature
was written. Classical Latin is what you learn when you learn Latin.
The pronunciation is fairly accurately known: see Classical Latin pronunciation for details.
In the Roman Empire, Latin was of course a living language. As such it continually changed. No-one has ever codified and frozen a living language: it is people's speech, not authorities' rulings, that determines how a language changes. There was a great flowering of literature at one time, then the language moved on, as they all do. However, such was the prestige of this so-called Golden Age that that style continued to be used not just one or two hundred years later (the Silver Age), but has continued to be the written form of Latin ever since.
So "Classical Latin" refers to both a period and a style. As a period, it was preceded by Old Latin and succeeded by Late Latin (or Vulgar Latin, see below). It's impossible to put dates on such changes because they're continuous, but let's stick a finger in the air and call it 200 BCE to 200 CE. Or for a narrow definition, 100 BCE to 100 CE.
In this period wrote Virgil, Cicero, Julius Caesar, Livy, Lucretius, Catullus, Ovid, Propertius, and Tibullus, in the Golden Age, followed 100 to 150 years later by the Silver Age of Tacitus, Suetonius, and Pliny. These authors and others set the literary standard that is associated with Classical Latin: their style remained long after the period when it was a spoken language.
In Rome, just as in the modern English-speaking world, there is a significant difference between the more formal and conservative language used in publications, and the language spoken in the street, home, and factory. Written English is still very close to that used 200 or even 300 years ago: authors like Jane Austen, Swift, or Fielding look merely old-fashioned, without the slight foreignness of Shakespeare before them. Yet our colloquial language has moved on a lot more. The same situation occurred in Rome and its empire.
During the Golden Age the spoken and literary languages were essentially the same, the difference being of course that Cicero and Caesar were great writers, skilled orators, and wrote their language in an ornate, convoluted style*.
In later years as this style became the fixed standard but the spoken language continued to change, substantial differences between writing and speech developed. This is called Vulgar Latin, from vulgus 'the common people'. The educated continued to use Classical Latin, and grammarians tried to get them to speak it as well as write it, but inexorably the bulk of the population used a language that was increasingly incompatible with the written text. (See the excellent write-up on Vulgar Latin for details of its development.)
All the Romance languages, descendants of Latin, descended from the spoken language, Vulgar Latin. It descended from the earlier spoken language, Classical Latin, and that from Old Latin, and that from Proto-Italic, and so on back indefinitely.
I have noticed however that an extraordinary story is bruited about, which I declare to be an urban legend: that the great authors like Cicero didn't speak Latin. One person infected with this meme said his Latin teacher had suggested they spoke a different language -- though he couldn't say what. A slightly more sophisticated version of this fiction is that they actually spoke Vulgar Latin, and that Classical Latin was an invention.
This confuses the high-quality literary expressions of individual writers with the underlying grammar, vocabulary, and pronunciation. The fact that Latin grammar is so ugly, irregular, and irrational is proof enough that it hasn't been tampered with: but linguists can also clearly trace the line of change from Proto-Indo-European to Classical Latin to Vulgar Latin and thence to French, Spanish, Italian, and the rest.
* evilrooster has pointed out that Caesar's style is not convoluted. True: his much-parodied style loads coordinated clauses together; it is long rather than deep.