The south-west 61% of Cyprus is controlled by Greek Cypriots and the north-east 39% is controlled by Turkish Cypriots.
Cyprus is in the Mediterranean Sea. South east of Greece, South of Turkey. Turkish Cypriots and Greek Cypriots live separately after the Turkish invasion in 1974. Turkish Cypriots live in the North while Greek Cypriots live in the South. The Greek Cypriots joined EU as the Republic of Cyprus in 2004. The North is not recognized as a country by any country besides Turkey.
The majority of Greek Cypriots identify as Greek Orthodox. Most Turkish Cypriots are adherents of Sunni Islam.
The majority of Cyprus is made of Greek Cypriots and a minority is made of Turkish Cypriots. There have also been an influx of forced migration from the Turkish mainland that has doubled the number of "Turkish Cypriots" even though these people have no Cypriot heritage and have been used in an attempt to change the population dynamics in Cyprus.
Greek Cypriots if born on Hellenic side or Turkish Cypriots on Turkish Side
They are called Cypriots, but usually prefer to be known by whether they are of Greek or Turkish origin. They would describe themselves as either Greek Cypriots or Turkish Cypriots.
They are both because Greek Cypriots sometimes speak Turkish and Turkish Cypriots sometimes speak Greek. But Turkish Cypriots always have a different accent to people from Turkey because they don't like to think of themselves as Turkish people, they think of themselves as Cypriot's. Greek Cypriots are the same.
The answer you are looking for is TURKEY, but the question improperly poses its premise. Greece is not in a conflict with Turkey over Cyprus. Turkey is in a conflict with the Greek Cypriots (the population of Cyprus which is ethnically Greek) over control of the island.
Cypriots
Cypriot. See http://en.wikipedia.org/wiki/Cyprus
The two ethnic groups are Greek Cypriots and Turkish Cypriots. Two other ethnic groups are Armenians and Maronites
The Cypriots travel wherever they please. They have airlines, ships, etc. The Cypriots are predominantly of Greek original, with a substantial portion being of Turkish origin, due to the island being conquered and occupied by the Ottoman Empire during its history. Then enter the British, and, when all is said and done with them, you end up with the current situation in Cyprus, and the breakaway Turkish Republic of Northern Cyprus, which was in response to the Greeks attempting a military coup of Cyprus in 1974 for the sake of Enosis - the annexation of Cyprus as part of Greece - preempted by attacks on the Turkish Cypriot population by nationalist Greek Cypriots under the direction of then-president Archbishop Makarios. But that's a long story there.