Learn Hardcore Russian: Старик улыбается прохожим каждый день. - The old man smiles at passersby every day.

The verb улыбаться (“to smile”) takes its object in the dative, not the accusative. In Russian, when you “smile at” someone, that someone goes into the dative case. Since прохожие (passersby) is plural, it becomes прохожим.

What does the -ся in улыбается mean?

The suffix -ся is a reflexive marker that turns the verb into an intransitive form. Улыбаться literally means “to smile oneself,” i.e. the subject (старик) is performing the action on its own face. Without -ся you’d have a transitive verb (rarely used) meaning “to make someone smile.”

Why is улыбается in the imperfective aspect instead of a perfective form?

Because the sentence describes a habitual action (“every day”). The imperfective verb улыбаться expresses repeated or ongoing actions. The perfective counterpart, улыбнуться, would imply a single completed smile (e.g. “he smiled once”).

What case is каждый день, and why do we use it here?

День is in the accusative singular because Russian often uses the accusative case for time expressions answering “how often” or “how long.” After каждый (“every”), the noun is still in accusative: каждый день, каждую неделю, каждый месяц.