The modern language started taking form about 200-300 years ago when Greece was ruled by the Ottoman Empire.
Modern Greek was divided into "Dhimotiki" ("demotic," "of the people", a term similar to "vernacular") and Katharevousa. Dhimotiki was the language of daily use, and the latter was an archaic form, used for official documents, literature, newscasting and other formal purposes. In the late twentieth century, it has been totally obsoleted and replaced by Dhimotiki.
Because of the long lasting domination under the Ottoman Empire, it's natural to assume that Modern Greek contains some vocabulary borrowed from Turkish. In later years, some French and Italian words were included.
A similar situation continues till now with English words, which in the last decades (like the borrowings from Turkish, Italian, and French), have been adapted to greek phonology and grammar. English loan words are found mainly in slang used by young people and in technical terminology.