With people who are techno savvy, I say that Bitcoin is the world's first truly peer-to-peer digital currency. That piques their interest.
For others, it's the concept of mining and how Bitcoin is created that is a challenge to explain.
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There's a new faucet in the middle of town. It's dispensing a metal called Bitcoin. This new metal is vastly superior to traditional forms of currency (dollars, pesos, yen, even gold) in many ways, so people in this town are collecting it to trade for things.
Alice has her bucket hooked to the faucet. She believes in the idea of this new kind of gold and her bucket is pretty nice. She gets a block of coins every 10 minutes, but she must keep her bucket securely fastened to the faucet so others cannot steal it.
Along comes Steve. He believes in this idea as well. He hooks up his bucket, which is the same as Alice's, and contributes his time and resources to watch it. Because the faucet only dispenses a limited amount of coins every 10 minutes, he must split his reward evenly with Alice.
Soon Harry, Fred, Egon & Spangler show up with their buckets. Soon, the whole town is mining for bitcoins using the same bucket and splitting the loot.
Now here comes Mike with his fancy bucket. Not only is it bigger, but it's faster at collecting bitcoins. It's also more flashy so to scare away would-be thieves. Mike hooks up to the faucet and collects a larger percentage of bitcoins because of his better bucket. Soon, Alice, Steve, Bob notice Mike's fancy bucket and higher profits. They and the others are now running out to upgrade their buckets.
Before long, bucket shops have sprung up all over town selling the newest, largest, and fanciest buckets. The more of the best, most powerful buckets that are hooked up to the faucet, the more secure the Bitcoin system is, and the more coins are released into the wild (to a point). Alice, Bob, Mike and the others have become like system administrators for Bitcoin, helping to secure it from pirates, marauders and thieves; and their buckets process and approve transactions from all other Bitcoin users, so long as they stay properly connected to the original faucet.
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...And so on. You really have to dumb it down for some people. And that means no mentioning computers, mathematics, economics or the internet. To really get them thinking, you could end by saying that the faucets are not all in one town, but all over the world, and each faucet is simply an internet connection.
This is interesting but has some flaws. I will re-write it when I get home to my computer (I'm typing on my phone now) because I think it has some useful ideas but is confusing in certain ways.