Italians celebrate birthdays in much the same way that most European and North American countries do. When it comes to children's birthday parties, there is often a spread of cakes, crisps and sweets - although, rather than having a bowl of Jolly Ranchers or fruit roll-ups, Italians might be more inclined to source some fritelle or cannoncini from their local patisserie.
The exchange of gifts is commonplace, but the playing of games like 'pass the parcel' less so.
When it comes to adult birthday parties, they are more likely to be celebrated enjoying drinks or a meal with close friends and relatives, or perhaps visiting a discoteca or club for drinks and dancing. As in many cultures, it is common for friends or guests to buy drinks for the person celebrating their birthday, as opposed to cultures where the host is expected to supply drinks for his guests.
There are a few different ways to wish someone 'a happy birthday', tanti auguri or buon compleanno being the most common examples. You can read a little bit more about that, and learn to sing happy birthday in Italian on this related blurtit answer!