In Higher Education, the word Greek most commonly means the language spoken around what is now called Greece from around 700-300 BCE. It is a much older version of Modern Greek - the language of Greece today, which is spoken in other parts of the world, mostly around the eastern Mediterranean. Modern Greek is mostly studied in the UK, like any other foreign language, for purposes of business and travel, and for communication generally. It is one of the languages of the European Community.
The language that this article is dealing about is called more precisely Ancient, or Classical, Greek. Like all human languages, Greek changed over time. Of the many other varieties that have been recognised, the most important in history was New Testament Greek (koine) - the language in which the New Testament of the Christian Bible was written. The language commonly used around the same time (or "from around 300 B.C. to 300 A.D." (OED) is also called Hellenistic Greek, although Hellenistic is used in different senses and to mean varying periods around that date. Greek is still the language of several Christian sects, or Churches, most notably the Greek Orthodox Church.
It is Classical Greek that is most commonly meant by dictionaries who give etymologies of words. In them, as in AWE, when 'Greek' is cited as of value in tracing the origins of words, it is Classical Greek that is meant - unless stated to the contrary.
Classical Greek was the second classical language at the heart of the curriculum of Higher Education in Western Europe from the 16th century to the 19th century. Before that, it had been the language of education in the Roman Empire, having great prestige. Indeed, it was very influential on the development of literary Latin. (This is rather like the importance, in turn, of Latin as the language of the educated in the British Empire.) Writing in Greek was regarded in the nineteenth century as the pinnacle of literary history. Together with Latin literature, it made up the branch of studies known as The Classics.
From the fall of Rome and the Western Roman Empire (5th-7th centuries CE), the knowledge of Greek effectively disappeared from Western Europe, although the writings of Aristotle, in particular, were widely translated and indeed revered in early Islam, where he was known simply as the Philosopher. A revival of Greek learning in western Europe came about on the capture of Byzantium, now called Istanbul, by the Ottomans in 1453, when many manuscripts were brought westwards and began to be read. This was a strong element in the shift in culture that we call the Renaissance, particularly the philosophy of humanism. In turn these contributed to the Reformation of the Christian religion.