Classical Latin is the language used by the principal exponents of that language in what is usually regarded as 'classical' Latin literature. Its origins within the framework of Indo-European are well understood, but there is much work still to be done on details.
Classical Latin is the Latin language of the Golden Age (broadly the 1st century BC), possibly extending to the Silver Age (broadly the 1st century AD).
What we call "Classical Latin" was, in fact, a highly stylized and polished written literary language selectively developed from early Latin, of which we have far fewer remains. The earliest Latin literature, such as Cato the Elder, Plautus and to some extent Lucretius, differs from the Latin of the "Golden Age" to some degree. The spoken Latin of the Roman Empire, especially from the second century forward, is generally called vulgar Latin. Vulgar Latin differed from classical Latin in its vocabulary and grammar, and as time passed it differed from the classical language in pronunciation as well.