The local Italian airlines are called Alitalia. As for other air companies, they are, for example, American Airlines, Lufthansa, Etihad Airways, British Airways, to name a few. Thus the best airline to fly to Italy is completely up to your preferences as the list of flight options to select from is very long regardless of seasonality.
Based on the city you depart from and arrive in there will usually be several daily or weekly departures to select from. In order to get a hold of the best airfare to Italy, it is recommended to book your flights in advance. The general rule is that the earlier you buy your tickets the cheaper they will be, moreover, you can save money by booking round-trip flights to Italy. Plus you'll have better availability and a broader choice of tickets.