Also known as Archaic Chinese (Bernhard Karlgren's term) and early Chinese. Chinese terms are shang4-gu3-yin1 and, traditionally, gu3-yin1 "ancient sounds".
Old Chinese refers mainly to the phonology, that is to say the sounds, of the Chinese language before the Han dynasty. Actual writings from this time are called Classical Chinese, not Old Chinese. The phonology of the period from the end of the Han through the Tang and up to the 10th century or so (the middle of the Song dynasty is now called Middle Chinese (Karlgren's Ancient Chinese). Everything after that is usually called (don't laugh) Modern Chinese.
Because Chinese is written in characters rather than an alphabet, writings of many different periods look superficially the same on the page, and anyone fully trained in Classical Chinese can read them. Unlike languages written in an alphabetic script, the shape of the words on the page tells nothing about how those words actually sounded when the text was written. For most writing, that makes little difference in the reader's ability to understand it, but in the case of poetry (which always rhymes in Chinese) people had begun to realize by about the Tang dynasty that something was going seriously wrong. Large quantities of ancient poetry did not rhyme properly, and that severely damaged its aesthetic appeal.
At first people solved the problem by substituting special pronunciations for individual rhyme-words that didn't rhyme with the rest of the poem. But by no later the Ming dynasty, scholars had begun to realize that ancient poetry failed to rhyme because the sounds of the language had changed. Philologists such as Chen Di (1541-1617), Gu Yanwu (1613-1682), and Jiang Yong (1681-1762) studied the rhyming of ancient poetry as a whole and proposed new systems of ancient pronunciation to account for it. A radically new method of study was discovered by Duan Yucai (1735-1815), who realized that the structure of the xiesheng characters themselves could be put into the equation; it was not necessary to look for rhyming uses of a given character, as long as it held phonological information in the form of a xiesheng element. Duan's discovery opened up about 90% of the characters to be studied within Old Chinese phonology, and in the 19th century a growing number of Chinese scholars built up the field.
Twentieth century scholars made several great breakthroughs. the Swedish sinologist Bernhard Karlgren (1889-1978) clothed the results of Duan Yucai and his successors in alphabetic form, so that the phonetics of Old Chinese could actually be visualized. The field began to take on a mixed Chinese-Western character, with scholars from Japan, France, Germany, Canada, the United States, and other countries, as well as China, taking part. Some students discovered likenesses between Old Chinese and Tibeto-Burman languages, leading to the Sino-Tibetan hypothesis - the theory that Chinese, Tibetan, Burmese, and many smaller languages all descended from a common ancestor. Others became interested in evidence of morphology in the old language, and still others began to notice traces of the old language in contemporary dialects.
We still do not really know very much about the pronunciation of Old Chinese, or its dialects, or the sociolinguistics of its living usage. But scholarship continues, never-ending.