Learn Russian: Здесь нельзя курить.

Russian often uses impersonal sentences for general rules. Instead of saying "you must not smoke" or "one must not smoke," the sentence is structured roughly as "here [it is] forbidden to smoke." There is no grammatical subject at all!

How does the word нельзя work with the verb?

The word нельзя means "it is forbidden" or "it is impossible." It is a special category of word that never changes its form. To say an action is forbidden, you simply put нельзя right before the infinitive (dictionary) form of the verb, which is курить (to smoke) in this case.

Could I change the word order to Нельзя курить здесь?

You would be understood, but Здесь нельзя курить sounds much more natural. In Russian, words that set the scene—like здесь (here) or сегодня (today)—usually go at the very beginning of the sentence.

Since the lesson is about reflexive verbs, why don't we use a reflexive verb here to say "smoking is forbidden"?

Even though reflexive verbs ending in -ся are sometimes used to express passive ideas, Russian has a specific, everyday way to talk about rules. For "allowed" or "forbidden," it is standard to use the fixed words можно or нельзя plus an infinitive, rather than trying to make a verb reflexive.