The basic metaphor
There is a worldwide, collective bulletin board where people can post two kinds of messages: transfers and discoveries. This log is shared and replicated (and secured) by hundreds of computers worldwide. It's public information and can't be easily blocked or modified without someone having access to every computer with a copy.
Transfers notes say "I, Jeff, give Kimberly 100 bitcoins". (Well, Jeff and Kimberly are account numbers, but you get the idea)
Discoveries say "I am posting a number which solves {some very very difficult well known mathematical problem} for which I am entitled to 10 bitcoins" (Where "I" is also an account number)
For various reasons, no one can forge either kind of message, or post it to the blog if it is not true. (Very complicated and well researched cryptography makes this possible)
Since the bulletin board is a log, and not a summary, every bitcoin account can be checked for balance by everyone else in the world by just looking up all the transfers in and out of it since the beginning of time (in 2009) You could do it with pencil and paper if you wanted to, just like in college accounting.
Why is this exciting for people?
First, it's anonymous cash. Since addresses are completely anonymous and easy to generate, a real person might have a thousand or more of them as places to which money can be sent and from which it can be spent. Of course you'd need to keep a lot of virtual checkbooks to do that, but computers are good at things like that. Since all the accounts are effectively anonymous, it means that it's not clear which "checks" in the ledger are between different people, and which ones are just a person transferring money between their accounts.
Second, the maximum number of bitcoins that can be "discovered" is finite: about 21,000,000. This means that there can never be "inflation" in the bitcoin world. No bank can make it worthless by printing more.
Third, since there are no "coins", only "balances", it can't be counterfeited. Anyone who posted a message saying "I transfer 100 bitcoins from account X to account Y" can be ignored if the account X had a balance under 100 coins. Thus, it cannot be counterfeited.
In summary: Anonymity makes it hard to track, tax and regulate. A medium which is proof against both inflation and counterfeiting makes it hard to devalue. These are things which make it very attractive to people.
So there's no downside?
Well, No. There are two potential downsides.
First, there is no country or bank backing up the value of the currency. It's literally worth exactly as much as the average user of the currency is willing to pay. That means its value can go up and down very rapidly in the short term. In the long term its inflation-proof nature will cause it to slowly increase in value over time.
Second, in the modern world, there's almost always some way to reverse a transaction. You can dispute a charge-card charge, you can cancel a check, you could even get a judge to order the vending machine company to give you your quarter back. With bitcoin, all transfers are absolutely and permanently final. There is no "undo" key. Some users see this as an advantage, but others see it as a disadvantage. The expression "caveat emptor" has never been embodied so completely in a medium of exchange than it has in bitcoin.
So there you have it. What it is, why it's exciting, and the downsides. Go forth in wisdom, and teach others!
