http://www.reddit.com/r/explainlikeimfive/comments/1ka962/eli5_how_bitcoins_work_mining_them_and_spending/
Quote
You and I both have a large sheet of paper. Both of our papers have two lines, reading:
Account 1: 5 bitcoins
Account 2: 2 bitcoins
You are account 2, I am account 1. Both of us also have a secret signature that we can verify to be authentic, but can't actually see what it is (this is public/private key cryptography, and is beyond 5 year olds) Let's say I want to send you 2 bitcoins. To do that, I take out a scrap of paper, and write
Transfer 2 Bitcoin from Account 1 to Account 2
--- Signed, Account 1's Secret Signature.
Then I write the following to the first big sheet of paper
Account 1: -2 Bitcoin
Account 2: +2 Bitcoin
And pass that scrap of paper to you, and everyone else using bitcoins. When you get that scrap of paper, you look at your own sheet to see if I have the 2 bitcoins to transfer from Account 1 to Account 2, check that the signature that was used to sign that note is really valid (matches Account 1) and if yes, modify your paper in the same way as mine. Owning that secret signature is what enables you to spend bitcoins, so me owning the secret signature for Account 1, and you owning the secret signature for Account 2, allows us to spend money from our accounts. In the end, the balance of each account is the beginning balance, plus all the changes we have added on later (so, 5-2=3 for Account 1, and 2+2=4 for Account 2).
Now, let's say I write a message asking to transfer 9 bitcoins from Account 1 to Account 2, and pass that message on to you. When you get it, you will check your sheet, and see that Account 1 doesn't have 9 bitcoins. Likewise, let's say that I send you a message saying I want to transfer 2 bitcoins from Account 5 to Account 2. Even if the secret signature for Account 5 is valid, when you look at your sheet, you'll see that Account 5 doesn't even exist. This is basically how Bitcoin works, and what prevent someone from creating counterfeit bitcoins, or just creating them out of thin air.
Now, regarding mining, basically, instead of you doing the work of writing those changes to the big sheet, you are delegating someone else to do it. You still look at the sheet to verify that accounts have the money they say they do, and you verify that the signatures are correct, but what Bitcoin miners do is they actually thoroughly verify the transfer messages, write that information down on the sheet, and then generate an EXTREMELY complicated and nearly impossible to forge signature to sign off on all changes to the paper. It's as if they take each completed sheet full of these transactions, and stick it into a glass safe, where everyone can look at the paper, but no one can alter or mess with it. This keeps others from going back and deleting old transfers, or fudging with the numbers in any way, so, that, for instance, I can't send you 2 bitcoins, and then go back, delete that transaction, and try to send you those 2 bitcoins again (double-spending). For the work of verifying and securing the papers, miners get to write a new account on the piece of paper with 25 new bitcoins in it, which from then on belongs to them. That's the only way a system allows for new bitcoins to be added to the sheets, and how Bitcoin gets distributed in the community.
Account 1: 5 bitcoins
Account 2: 2 bitcoins
You are account 2, I am account 1. Both of us also have a secret signature that we can verify to be authentic, but can't actually see what it is (this is public/private key cryptography, and is beyond 5 year olds) Let's say I want to send you 2 bitcoins. To do that, I take out a scrap of paper, and write
Transfer 2 Bitcoin from Account 1 to Account 2
--- Signed, Account 1's Secret Signature.
Then I write the following to the first big sheet of paper
Account 1: -2 Bitcoin
Account 2: +2 Bitcoin
And pass that scrap of paper to you, and everyone else using bitcoins. When you get that scrap of paper, you look at your own sheet to see if I have the 2 bitcoins to transfer from Account 1 to Account 2, check that the signature that was used to sign that note is really valid (matches Account 1) and if yes, modify your paper in the same way as mine. Owning that secret signature is what enables you to spend bitcoins, so me owning the secret signature for Account 1, and you owning the secret signature for Account 2, allows us to spend money from our accounts. In the end, the balance of each account is the beginning balance, plus all the changes we have added on later (so, 5-2=3 for Account 1, and 2+2=4 for Account 2).
Now, let's say I write a message asking to transfer 9 bitcoins from Account 1 to Account 2, and pass that message on to you. When you get it, you will check your sheet, and see that Account 1 doesn't have 9 bitcoins. Likewise, let's say that I send you a message saying I want to transfer 2 bitcoins from Account 5 to Account 2. Even if the secret signature for Account 5 is valid, when you look at your sheet, you'll see that Account 5 doesn't even exist. This is basically how Bitcoin works, and what prevent someone from creating counterfeit bitcoins, or just creating them out of thin air.
Now, regarding mining, basically, instead of you doing the work of writing those changes to the big sheet, you are delegating someone else to do it. You still look at the sheet to verify that accounts have the money they say they do, and you verify that the signatures are correct, but what Bitcoin miners do is they actually thoroughly verify the transfer messages, write that information down on the sheet, and then generate an EXTREMELY complicated and nearly impossible to forge signature to sign off on all changes to the paper. It's as if they take each completed sheet full of these transactions, and stick it into a glass safe, where everyone can look at the paper, but no one can alter or mess with it. This keeps others from going back and deleting old transfers, or fudging with the numbers in any way, so, that, for instance, I can't send you 2 bitcoins, and then go back, delete that transaction, and try to send you those 2 bitcoins again (double-spending). For the work of verifying and securing the papers, miners get to write a new account on the piece of paper with 25 new bitcoins in it, which from then on belongs to them. That's the only way a system allows for new bitcoins to be added to the sheets, and how Bitcoin gets distributed in the community.