The Italian equivalent to the English phrase 'Happy Birthday' is 'Buon Compleanno'.
The phrase can be broken down into two words, the first- buon is a shortened version of buono, the final 'o' being dropped as a form Apocope (a linguistic convention that the Italian language adheres to).
The meaning of buono is 'good', or 'beneficial' and is often used in well-wishing as in the famous phrase 'buon appetito' which is used to wish a diner a 'good' appetite.
Compleanno is a compound word, with the comple- part coming from the root word compiere meaning to complete. The second part -anno means year.
Another phrase you can use to wish someone a happy birthday is 'Tanti Auguri' meaning 'Many Regards or wishes'. It is this phrase that is sung along to the tune of 'Good Morning To All' (written by Mildred & Patty Hill in 1893) as a tribute to the person who is celebrating their birthday, the English equivalent of the 'Happy Birthday' song.
It's interesting to note that the expression 'Auguri' is by no means limited to birthdays, and can be used to express your best wishes to anyone for almost any celebratory occasion including New Year's Day, Christmas Day, and the arrival of a new baby to name a few.