More than congratulations (congratulazioni) in Italy, people give auguri, or well wishes. In other words, it’s a way to wish someone well in the future, or a way to say ‘all the best.’ You use it to wish someone a happy birthday (in addition to the less used, buon compleanno). You can also say it to someone who recently graduated, got married, or completed some difficult task.
‘Auguri di buon anno!’ – Happy new year!
‘E’ il tuo compleanno, Pietro?! Auguri!’ – Peter, it’s your birthday?! Best wishes!
‘Auguri!’ – Mazal tov!
Auguri is actually the plural form of the masculine noun, augurio. Augurio is a wish, or a hope expressed which can be positive or negative for one’s future. Therefore, it can also mean a bad omen, or an ill wish directed toward someone. However, when spoken in the plural, auguri is used for auspicious occasions.
Augurio stems from the Latin, augurium – the foreseeing, or the divination of the future through song, or through the flight of birds (for the Romans).
Side-note: Wishing someone auguri for his/her birthday before the actual day, or before an exam is quite unappreciated, and considered superstitiously bad luck.