Learn Russian: Один из студентов опаздывает каждый день.

Here it means 'one'. You can easily tell because it is followed by из (out of/from). The phrase один из always means 'one of'. If you wanted to say a student is late alone, you would instead say студент опаздывает один (or одна for a female student).

Why is the word for students (студентов) in the genitive plural?

The preposition из always requires the genitive case. Because we are saying 'one out of the students', we need the plural genitive form of студент.

Why use the imperfective verb опаздывает instead of a perfective verb for being late?

The clue is каждый день (every day). Because this is a regular, repeating habit, Russian requires the imperfective aspect. A perfective verb like опоздает would mean someone will be late just one specific time.

What case is каждый день in, and why?

It is in the accusative case. Time expressions that describe frequency (like 'every day', 'every week', or 'every year') take the accusative case without any preposition. For masculine inanimate nouns like день, the accusative form looks identical to the nominative.