Just like in Paris, New-York or London, Dubai his full of great and original restaurants. There are restaurants for everyone, familly restaurants as well as little european style bistrots. Some of the world's most renowned chefs work in Dubai. 

Local arabic food can be found in almost every restaurants. It's a rich and very diverse cuisine. Every tourists should at least try it once. You will learn new recipes, and taste totally new and exotic dishes. Going to Dubai is a great way to discover, while there, all the richness of the Arabic gastronomy.  You will love it!

Here is a list of almost all restaurant serving traditionnal Arabic food in Dubai:

http://www.dubai-eating.com/cuisines/arabic.asp