Learn Hardcore Russian: Нужно ценить каждый момент жизни. - You should appreciate every moment of life.

Нужно is an impersonal modal predicate expressing necessity. Literally it means “it is necessary,” so the full idea is “It is necessary to appreciate every moment of life.” In natural English we’d say, “You must appreciate every moment of life.”

Why is ценить in the infinitive form instead of a conjugated verb?

After impersonal necessity words like нужно or надо, Russian requires the action verb to stay in the infinitive. The infinitive ценить simply names the action that must be done (“to appreciate”).

Why is there no explicit subject like “we” or “you” in this sentence?

Because the construction with нужно is impersonal. There is no grammatical subject—нужно applies generally to anyone. In English we often add “it” or “you” for convenience, but in Russian you leave it out.

What case is каждый момент, and why does it look like the nominative?

Каждый is nominative masculine singular to agree with момент. As the direct object of ценить, момент should be in the accusative case. However, since момент is an inanimate masculine noun, its accusative form is identical to its nominative. Hence каждый момент serves as the object.