Learn French: Mon rendez-vous est après-demain. - My appointment is the day after tomorrow.

You would ask, Quand est ton rendez-vous ?. You use quand (when) because the answer is a general day (après-demain). If you wanted to know the exact time on the clock, you would instead ask, À quelle heure est ton rendez-vous ? (At what time is your appointment?).

Is après-demain literally "after tomorrow"?

Yes! In French, demain means "tomorrow" and après means "after". Put them together with a hyphen, and you get après-demain, which is the exact equivalent of the English phrase "the day after tomorrow".

Does rendez-vous mean a doctor's appointment, or a romantic date?

It can mean both! A rendez-vous is simply any pre-arranged meeting. You use the exact same word for a trip to the dentist, a business meeting, or a romantic date. The context will tell you which one it is.

Does the "s" at the end of rendez-vous mean I have multiple appointments?

No, rendez-vous is strictly singular in this sentence. We know this because of the singular word mon (my) and the verb est (is). The word rendez-vous just naturally ends in an "s" even when there is only one. To make it plural, the spelling stays exactly the same, but the words around it change, like in mes rendez-vous sont (my appointments are).