Mezedes (little dishes)
In Cyprus like many of the Greek Islands and Mainland, if you are offered a drink you will always be given a little something to eat as well (and vice-versa). A good traditional household inviting you in for company would almost always offer you a drink and a small selection of finger/snack food (as drink shouldn't be taken without food). This is almost certainly the origin of the Meze as it has become known in modern restaurants.
Restuarants have taken this idea and developed it into an experience not to be missed if you want to try their wares. You will usually be given things like Olives, Pitta, Humous (a tahini flavoured chickpea paste), Tahini (a sesame paste), taramasalata (fish spread) and salad. These are brought out for you to pick at, like a Buffet in the Uk or Tapas in Spain. More often than not you will also be treated to a portion of several of the house main meals like "stifado" (a rich stew with baby onions), "kleftiko" ( a slow oven roasted lamb dish), "Afelia" (Pork with redwine, vinegar and corriander) as well as oven roasted potatoes, pasta and any other things that they might like to showcase on the meze.
Different restaurants will have variations, some will do vegetarian and fish versions, some will have more courses, some less... you pay's your money, you takes your choice. You will find its a very good way of being introduced to the local foods and being able to see what that restuarant is like. From this you may find you don't like one thing but another sharing might love it, you eat a bit more of one thing, they eat a bit more of another etc.... By the end of the "meal" (I use the word meal lighlty as its more like a banquet) you will have made some choices about what you might like to try again (even if its at a different restuarant next time)...
Steve