The post-classical coinages are by far the most numerous of these.
Since the living Greek and English languages were not in direct contact until modern times, borrowings were necessarily indirect, coming either through Latin (through texts or various vernaculars), or from Ancient Greek texts, not the living spoken language.
Some Greek words were borrowed into Latin and its descendants, the Romance languages. English often received these words from French. Their phonetic and orthographic form has sometimes changed considerably. For instance, place was borrowed both by Old English and by French from Latin platea, itself borrowed from Greek πλατεία (ὁδός) 'broad (street)'; the Italian piazza and Spanish plaza have the same origin, and have been borrowed into English in parallel. The word olive comes through the Romance from the Latin word olīva, which in turn comes from the Greek ἐλαίϝᾱ (elaíwā). A later Greek word, βούτυρον (boútȳron) becomes Latin butyrum and eventually English 'butter'. A large group of early borrowings, again transmitted first through Latin, then through various vernaculars, comes from Christian vocabulary: chair << καθέδρα (cf. 'cathedra'), bishop << ἐπίσκοπος (epískopos 'overseer'), priest << πρεσβύτερος (presbýteros 'elder'), and church < Old English cirice, circe < probably κυριακή (kȳriakḗ 'lord's '). In some cases, the orthography of these words was later changed to reflect the Greek – and Latin – spelling: e.g., quire was respelled as choir in the 17th century. Sometimes this was done incorrectly: ache is from a Germanic root; the spelling ache reflects Samuel Johnson's incorrect etymology from Greek ἄχος.
Many more words were borrowed by scholars writing in Medieval and Renaissance Latin. Some words were borrowed in essentially their original meaning, often transmitted through classical Latin: topic, type, physics, iambic, eta, necromancy, cosmopolite. A few result from scribal errors: encyclopedia < ἐγκύκλιος παιδεία 'the circle of learning', not a compound in Greek; acne (skin condition) < erroneous ἀκνή < ἀκμή 'high point, acme'. Some kept their Latin form, e.g., podium < πόδιον.
Others were borrowed unchanged as technical terms, but with specific, novel meanings: telescope < τηλεσκόπος 'far-seeing' refers to an optical instrument for seeing far away rather than a person who can see far into the distance; phlogiston < φλογιστόν 'burnt thing' is a supposed fire-making potential rather than something which has been burned, or can be burned; bacterium < βακτήριον 'stick (diminutive)' is a kind of microorganism rather than a small stick or staff. This also applies to combining forms used in neologisms: -cyte or cyto- < κύτος 'container' refers to biological cells, not arbitrary containers.
But by far the largest Greek contribution to English vocabulary is the huge number of scientific, medical, and technical neologisms that have been coined by compounding Greek roots and affixes to produce novel words which never existed in the Greek language: utopia (1516, οὐ 'not' + τόπος 'place'), zoology (1669, ζῷον + λογία), hydrodynamics (1738, ὕδωρ + δυναμικός), photography (1834, φῶς + γραφικός), oocyte (1895, ᾠόν + κύτος), helicobacter (1989, ἕλιξ + βακτήριον). Such terms are coined in all the European languages, and spread to the others freely—including to Modern Greek. Traditionally, these coinages were constructed using only Greek morphemes, e.g., metamathematics, but increasingly, Greek, Latin, and other morphemes are combined, as in television (Greek τῆλε + Latin vision), metalinguistic (Greek μετά + Latin lingua + Greek -ιστής + Greek -ικος), and garbology (English garbage + Greek -ολογία). These hybrid words were formerly considered to be 'barbarisms'.
Many Greek affixes such as anti- and -ic have become productive in English, combining with arbitrary English words: antichoice, Fascistic.