A couple things:
First, the charm is meant to protect a secret, like it has been stated in earlier posts. I would guess this means, like the Grimmauld Place secret, that only one person, the secret keeper, is privy to this secret and only through that person can the secret be told, I.E. Dumbledore's note with the address on it.
Secondly, I would think it works something like an unbreakable vow, where the only people present at the time of casting would have been, in the Potter's case, Lily, James and Peter (and baby Harry, of course).
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I would believe they just had to transfer the charm to another person in order to make it active again. Just like Sirius did to make Peter the Secret-Keeper.
I disagree with this, based on what Sirius says. He tells Harry that he convinced James and Lily to switch to Peter as their secret keeper at the last minute, meaning that Sirius never
was their secret keeper. Therefore, according to what I've hypothesized eariler, he wasn't there for the actual casting of the charm and Peter therefore had to tell Sirius where the Potters were hidden.
Also, I personally think the secret was the location of their cottage in Godric's Hollow, not Lily and James themselves, cause it would be really weird to see blocks of wood magically splitting themselves. ;-P
I would also suspect that there would have been more than just the Fidelius Charm protecting them - or at least I would hope so! I would have thought that Dumbledore, Sirius, Remus, or even the Potters themselves would have cast charms around the house, or maybe even thier village, to prevent unwanted wizards from wandering through town and that Voldemort had been told about these charms by Peter so he knew he had to break through them (an easy task for a wizard with forewarning about them).