Learn Russian: Ему нужен билет.

In English, "he" is the subject who needs something. But in Russian, the construction for "need" is literally "to someone is necessary something". The person who experiences the need takes the Dative case (Ему = to him), not the Nominative case (Он = he).

Since the lesson is about modal verbs like хотеть (to want), is нужен also a verb?

No, нужен is actually a short adjective meaning "necessary". This is why it behaves differently than the verbs you are learning. Verbs like хотеть change endings based on who is doing the action (I want, he wants), but нужен changes based on the gender of the thing being needed.

If "he" is the one who needs the ticket, why isn't билет (ticket) in the Accusative case as a direct object?

Because the literal translation is "A ticket is necessary to him", the ticket is actually the grammatical subject of the sentence! That is why билет stays in its base dictionary form (the Nominative case).

Would нужен change if he needed something else, like a car (машина)?

Yes! Because нужен is an adjective describing the thing needed, it must match that thing's gender. Билет is masculine, so we use the masculine form нужен. If he needed a car (a feminine word), it would change to the feminine form: Ему нужна машина.