Koine Greek was the form of the language that was spoken in the wider Hellenistic World from 300 BC to about 600 AD. The Hellenistic World was made up of the regions to which Greek culture was spread following the conquests of Alexander the Great, and included parts of the Near East and Egypt. The New Testament was written in Koine, also called Biblical Greek in this context.
Koine Greek is the ancestor of modern Greek.
See also Ancient Greek names.