Curiosity is the urge you feel to know more about something. If you find a diary in a coffee shop, curiosity will make you want to look inside — but respect for the writer's privacy may convince you not to.
Curiosity is the state of being curious: inquisitive, wondering, ready to poke around and figure something out. The word used to mean "very, very careful," and only in the last few hundred years turned into a word expressing the desire to know more. Maybe that's because when you're poking around in secret business, you need to be very, very careful. We also call weird things curiosities, like a phone shaped like a duck, or a shark in a jar.