Learn French: Nous avons besoin de vacances.

While vouloir means to want, this sentence means we need a holiday. In French, to say you need something, you use the expression avoir besoin de (literally: to have need of). If you wanted to say we want a holiday using the current lesson's verb, you would say Nous voulons des vacances.

Why is it de vacances and not des vacances since the word is plural?

The expression is built with de (avoir besoin de). When the preposition de is followed by the plural word des (some), they merge and shrink down to just de. So, besoin de plus des vacances simply becomes besoin de vacances.

Why is vacances plural here when the English translation just says a holiday?

In French, the word for holiday or vacation is almost always used in the plural form: les vacances. Even if you are just talking about one single trip or one break from work, you still use the plural word.