There are too many restaurants to count in and around Paphos and there is huge variety. For local food, there are tavernas, which serve traditional Cypriot food - kleftico (clay baked lamb which just falls off the bone), kebabs (called souvlaki in Cyprus), stifado (a beef stew) and halloumi cheese to name just a few local delicacies. To sample the full range order a meze - lots of little dishes will come to your table enabling you to try all sorts of different things but possibly not to walk again if you eat it all!
In Paphos itself you will find not only local food but also pizza, Japanese, Indian, Chinese, Thai, ribs and pretty much anything else you can think of.
All our Cyprus villas have restaurants close by.