One of the issues that Bitcoin suffers from the most is a lack of a concise explanation that people can latch on to. I would like to propose a straightforward analogy that may help introduce individuals to the way in which Bitcoin works; the actual mechanism. It is comprised of three items: a book, a pen, and ink.
1) The book
All of Bitcoin centers around an enormous book called the "blockchain" that keeps a record of every single Bitcoin transaction that has ever occurred. This is why Bitcoins are nearly impossible to counterfeit, because every time you spend some, your request is checked against this huge ledger.
People called "miners" surround this book, and do the job of verifying transactions. For some reason I like to imagine them wearing hooded cloaks. Every 10 minutes a lucky miner is selected from the group, and is allowed to approach the book, check everybody's work, sign his (or her) name in the corner, and then turn the page (or block). On the next page is a reward of 25 fresh Bitcoins that he gets to keep before returning to his spot. Upon returning, he shows his fellow miners his work so that everybody can agree it was valid.
2) The pen
In between page turns, anybody can make a transaction by writing in the book. But to make a mark, we need a pen, something that can interact with the page.
The pen is analogous to pieces of software we call "clients" or "wallets" and they can come in all shapes and sizes. Some are old fashioned fountain pens, whereas others are like souped up versions of those multi-color pens with six cartridges (you know the kind I mean). "Wallets" can be either kept on your personal computer or be hosted by a website, and today they are getting more and more sophisticated - think battery powered with booby traps.
But they are all just versions of the same thing, something that can write in the book. If you leave your pen lying around on your desk, somebody might pick it up and steal your Bitcoins. So it probably makes sense to keep this pen in a safe (encrypt it with a password) where only you have access to it.
3) The ink
The real magic isn't in your pen at all, but rather in the ink it carries (made up of your "addresses" and their "private keys"). There is an almost infinite number of ink colors in the Bitcoin world, and whenever you make a transaction, you want to make your own unique mix to prove that it was you.
The trick is to be very careful about keeping your ink secure. If it is stolen, then it can be used to make a transaction with your Bitcoins. Similarly to how you protect your pen, you can keep your ink locked up in a safe. And to ensure you don't lose access to your ink, you may also want to make some additional copies of it.
For instance, you could smear a little bit on a piece of paper. Now you have a paper wallet, which you can refer to later to remind you of your ink color. Or if you are really clever, you can memorize the recipe that you used to derive the ink; this is called a brain wallet. Then nobody can steal your coins without getting you to physically confess the information. When it's time for you to send your Bitcoins, you simply cook up a batch of your special ink, load the cartridge into your pen, walk up to the book and make an entry.
On your way back, you probably will want to chuck your empty ink cartridge in the bin. You can't be too careful these days. Better to just choose a different color ink next time.
Feel free to share your own Bitcoin analogies!