As a language name this word is uncountable. That means no a in front of it and no -s on the end: speak Hungarian, learn Hungarian, study Hungarian. To say that something is written or said in that language, use in Hungarian. Note the difference from the person noun: a Hungarian is a person, while Hungarian on its own is the language. Language names keep the capital letter too. One thing learners notice quickly is how much work the endings on a Hungarian noun do.

This is the country label in adjective form, built from the country name plus -an. It goes in front of almost anything that can be tied to Hungary: Hungarian food, Hungarian music, a Hungarian passport, a Hungarian friend. It can also point at the language: a Hungarian word, a Hungarian newspaper. The capital H stays whatever position the word sits in. Country adjectives like this one do not usually take more or most, because a thing is either from there or not. The same -an ending builds many other country adjectives, so the pattern is worth holding on to.

Here the same word stands on its own as a noun for a person. It is countable, so it needs a or the in the singular and takes -s in the plural: a Hungarian, two Hungarians, the Hungarians. English gives you two ways to say where someone is from. He is Hungarian uses the adjective; He is a Hungarian uses this noun. Both are used, and the adjective form is the more common one in everyday talk. The capital H stays in every case.