Learn Croatian: Živjeli sto godina! - May you live a hundred years!

This sentence uses the optative mood to express a strong wish or toast. In Croatian, you form this by using the bare l-participle (the verb form usually used to build the past tense) all by itself, dropping the auxiliary verb entirely. That is why it is just živjeli instead of ste živjeli or biste živjeli.

Who is živjeli addressed to here?

Živjeli is the masculine or mixed plural form. You use it when toasting a group of people, or when addressing a single person formally using the polite Vi. If you were addressing a single male friend, you would say živio, and for a female friend, živjela.

What case is godina in here, and why?

It is in the genitive plural. After numbers from five onwards (including sto), Croatian requires the following noun to take the genitive plural. The genitive plural of godina (year) happens to be spelled exactly like its nominative singular form, though the vowels are pronounced longer.

How common is this phrase in everyday Croatian?

Very common! While the full sentence Živjeli sto godina! is often used as a birthday wish or a heartfelt toast, the single word Živjeli! is the standard way to say "Cheers!" when clinking glasses in Croatia.