What language do people in Cyprus speak?

Eutychius Kaimakkamis’ is the most complete answer; I’ll only add:

• The status of Standard Greek vs Cypriot Greek is a diglossia, and it’s a much more clear-cut instance of diglossia than what was going on in Greece in the 20th century.
• Cypriot Turkish (Cypriot Turkish, Kıbrıslıca) has some clear typological affinities with Cypriot Greek. For instance, they share VSO, as opposed to Standard Turkish’s SOV and Standard Greek’s SVO.
• According to Ibn Hazm again, and Cypriot Arabic , Cypriot Arabic is even harder to understand for a speaker of Levantine Arabic than Nigerian Arabic: “It’s pretty difficult to even read, perhaps on par with the basilect of some English-based creoles.” He thinks there is Aramaic influence there.
• The links in my 2009 blog article about Cypriot Arabic have expired; but the community is now using the Roman alphabet, with the main writer in the language (the local priest) reluctantly abandoning the Greek script.
• Leontios Machairas in his 15th century chronicle of Cyprus famously described the triglossia of his time: French, Syriac, and Greek. “And because in this world there are two natural masters, one temporal and the other spiritual, this little island had the patriarch of the great Antioch, before the Latins took it over. For it was useful to know universal Greek in order to send petitions to the king, and correct Syriac. And it was in this way that the children were taught until the Lusignans took the place, and since then we started learning French and have barbarised Greek as it is today, when we write both French and Greek so that no one in the world knows what we speak.”
• Romani languages are always left out, and until I popped over to Languages of Cyprus, I did not know about Kurbet language: a Para-Romani based on Cypriot Turkish.