Learn Italian: I tuoi progetti e le nostre vacanze sono importanti. - Your projects and our vacations are important.

In Italian, possessive adjectives almost always need a definite article (like "the") in front of them. You literally say "the your projects" (i tuoi progetti) and "the our vacations" (le nostre vacanze). The main exception to this rule is with singular family members, which do not take the article.

How do I know which article and ending to use for the possessives?

They must match the gender and number of the thing being owned, not the owner! Progetti (projects) is masculine plural, so it takes the masculine plural article i and possessive tuoi. Vacanze (vacations) is feminine plural, so it takes the feminine plural article le and possessive nostre.

In English we often say "our vacation" (singular). Why is vacanze plural here?

While the singular word vacanza exists, Italians almost always use the plural le vacanze to talk about a holiday, a trip, or time off from work or school. So "our vacation" translates naturally to the plural le nostre vacanze.

Progetti is masculine and vacanze is feminine. How does the adjective importanti agree with both?