Ayia Napa has a wide range of restaurants covering everything from traditional Cypriot meze to international fare. The harbour area is the best spot for fresh seafood -- several tavernas there serve multi-course fish meze that is worth the time and the price. The Square and Nissi Avenue have more casual options, including fast food and international chains.
For a more local experience, look for places that serve kleftiko (slow-baked lamb) or halloumi straight off the grill. Portions tend to be generous and prices are reasonable by European standards.