What's the difference between Passato Prossimo and Passato Remoto?

It's a very complex question, but the broad answer boils down to this: it's your attitude about the past. If you think that a past event is connected to the present, or in any way important for you right now, you use Passato Prossimo. If it's unconnected, and you're indifferent about it, you use Passato Remoto. A good, if slightly morbid example of that difference can be the following:
E' morto il mio nonno. - My grandpa died (subtext: it's recent, and I'm sad about it) - Passato Prossimo
Il 1821 mori' Napoleone. - Napoleon died in 1821 (subtext: I don't feel about it either way) - Passato Remoto
To sum up, it's you and your feelings about the past that dictate which tense to use!