Ore is more or less the "o'clock" of the Italian language. It also means "hour" as in "it takes 3 hours to get there" (Ci vuole 3 ore per andare lì). It can also mean "now" : "Quando se non ora ?" ("When if not now?") and is used in the expression "Non vedo l'ora [di ...]" : "I can't wait [to ...]". When you talk about the time on a clock, always use ore. Tempo is more abstract, it could mean "weather" (Che bel tempo !), the time it takes to do something (Quanto tempo ci vuole per arrivare ?), "epoch" : (Inquesti tempi, la gente non faceva così) ("In those times, people didn't do that"), and is used in this expression : "Poco tempo fa", "Some time ago".

That is also my understanding, but I am not in a position to confirm it.