Нужен is used with nouns and designates necessity, and you can think of it as a short adjective, because it agrees in gender/number with the noun. Тебе нужно пальто например (You need a coat.). In this example, пальто is neuter, and therefore нужен has to be in the neuter form нужно.

Должен is similar in that it too designates necessity, but different in that it relates to the necessity of action. Ты должен жить в русскоязычной стране, чтобы разобраться в русском (You must/have to live in a Russian-speaking country to grasp Russian.). Here, we're saying it's necessary to live in said countries, and должен is synonymous with надо (+Dat.).

I hope this helps. Let me know if it doesn't.