Learn Hardcore Hungarian: Ez egy népszerű étterem.

egy is the indefinite article, equivalent to a or an in English. In Hungarian, when identifying or introducing one item as part of a category, you almost always include egy. Omitting it, as in Ez népszerű étterem, would sound ungrammatical or unnatural.

Where is the verb “to be”? Why isn’t van used here?

In Hungarian, the present-tense form of van (the verb “to be”) is usually omitted in simple identification sentences. You say Ez egy népszerű étterem instead of Ez egy népszerű étterem van; the existence of the verb is understood implicitly.

Does népszerű change its form to agree with the noun?

No. Hungarian adjectives are invariable: they do not change for gender, number, or case. So népszerű remains the same whether the noun is singular or plural, masculine or feminine.

Why does étterem have no ending? What case is it in?

Here étterem is a predicate noun in the nominative case, which has no suffix. It simply names what ez (this) is. You only add endings when you need other cases (e.g. locative étteremben, accusative éttermet, etc.).

How would you say “a very popular restaurant”?

You insert a degree adverb like nagyon before the adjective:
Ez egy nagyon népszerű étterem.

How would you say “the popular restaurant” (using the definite article)?

Replace egy (a/an) with the definite article a (or az before vowels):
Ez a népszerű étterem.

How is népszerű pronounced? Where is the stress and which vowels are long?

Stress in Hungarian always falls on the first syllable: NÉP-sze-rű.
The letters é and ű are long vowels, so you hold them slightly longer than short vowels.

What’s the difference between ez and az?

ez = “this” (refers to something close to the speaker or just introduced)
az = “that” (refers to something farther away from the speaker or already known)
So if you wanted to point to a more distant restaurant you could say Az egy népszerű étterem, meaning “That is a popular restaurant.”