Classical Japanese, also known as Literary Japanese (Japanese: 文語 bungo), is, strictly speaking, the language of Japan's Heian Period (794-1185), which was the first time Japanese authors began to write literature in their native language rather than in Chinese. It is thus the language of such famous works as Lady Murasaki's Tale of Genji and the Pillow Book of Sei Shonagon.
But despite the fact that spoken Japanese continued to evolve over the centuries since then, Classical Japanese came to be seen as the "correct" form of Japanese and remained in constant use as the written language of the educated elite and the literati of Japan all the way up until the end of World War II. Until the late Edo Period (1603-1868), virtually all Japanese literature of any kind was written in Classical Japanese. Even after most authors began writing literature in the vernacular in the Meiji Period (1868-1912), Classical Japanese remained in use in newspapers and government documents.
Compared to Modern Japanese, Classical Japanese had a much more complex verbal morphology, including nine classes of verbs (as compared to four in Modern Japanese), and a dizzying array of inflecting suffixes to mark aspect and tense and even the speaker's emotional state. Classical Japanese also had a slightly different orthographic system, which more accurately reflected the way Japanese words were pronounced 1,000 years ago.
Learning Classical Japanese is a must for anyone wishing to seriously study Japanese literature or history prior to World War II. Even today it is still occasionally used in traditional forms of writing such as the tanka poem, and by scholars writing Japanese glosses on Classical Chinese texts.