In Italy, by law everyone (over 18 years old) is required to carry at all times an official identity document. And you often have to show it - like when you check into a hotel. You can always carry around your passport if you like, but it is (relatively) bulky and can be quite difficult to replace.
So the solution is to get an Italian identity card, or "carta di identita'", which looks like this (sorry for the image quality, but you don't want to be there all day uploading a picture...)
For Italian citizens it can be used to travel within the European community, for non-Europeans it is a valid form of identification within Italy (also for boarding national flights). But it is not valid for travel or identification outside of Italy.
You can apply for this at the local "comune" at the resident registration office "ufficio anagrafe".
Special Note: It is against the law to laminate the carta d'identita'. If you woul dlike to protect it, you can buy a plastic cover for it at any local tobacco shop.