Separated in two parts, but both of them have some foods that can be found and even pronounced as traditional in their neightbor countries, Greece and Turkey. There are however, and some that are 100% Cyprus traditional foods and can't be seen anywhere else.
Halloumi - one of the very typical local meals, that can't really be found anywhere in its neightbor countiries. I think, I've seen in only one restaurant in Greece in my lifetime of traveling. Halloumi is a cheese made of a mixture of goat's and sheep's milk. Can be very easily grilled or fried and it's often served with salads, but you can have it as an appetizer as well.
Makaronia tou fournou -oven-baked macaroni in the oven also known as Pastitsio and can be seen in Greece, Turkey, as well as in my home country, Bulgaria. It's a very delicious pasta meal, if you're into Italian food you'll like this as well, I guarantee it.
Koupepia and gemista - is a family of stuff dishes similar do dolma. Common vegetables to stuff include tomato, pepper and eggplant. Cabbage rolls and dishes of stuffed grape leaves are also very popular, and these are sometimes also called sarma and can be fourd in Greece, Turkey and Bulgaria as well.
Souvlakia - traditional food not only for Cyprus, but for its neightbors as well. Souvlakia is typically made with chicken or pork meat, and in Turkey can be found with lamb. Consisting of small pieces of meat and sometimes vegetables, grilled on a skewer. It can be served with pita bread, fried potatoes and sauces, but the souvlaki itself is eaten on its own, with the side dishes eaten subsequently. Can rarely be found with beef or fish as well.
Souvla - a very popular meal, it consists of large pieces of meat cooked on a long skewer over a charcoal barbecue. Typically souvla is made from the neck and shoulder of lamb, pork or chicken meat and it's cut to the bone into chunks about the size of a medium onion.
Koupes - a popular dish in Middle Eastern cuisine and can be found in many countries such as Lebanon, Syria, Jordan, Egypt, Iraq Turkey and others. Made of bulgur, minced onions, and finely ground lean beef, lamb, goat, or camel meat with Middle Eastern spices.
Meze - a selection of small dishes served as appetizers in parts of the Middle East and the Balkans. It can and it is in many countries, especially in the Balkans used not only as an appetizer but as a snack while drinking alcohol.
Kleftiko - literally meaning "in the style of the Klephts". A slow-baked lamb meat on the bone, marinated in garlic, originally cooked in a pit oven. Legend says that Klephts, bandits of the countryside who did not have flocks of their own, would steal lambs and goats to cook the meat in a sealed pit to avoid the smoke being seen.