1) Start with cash
Everyone understands cash so start here. Cash can be stored in a wallet and exchanged in person for goods or services. Everyone accepts cash because we all believe it holds value. That value is derived from the fact that everyone accepts it, and that it is government backed. Your cash has not been backed by gold for many years, but we all trust in the economy and we all use it to buy things.
2) Now explain how banks store cash
Gone are the times where banks stored large volumes of physical cash in their vaults. Banks now hold electronic money. Only ~5% of money is actually printed on physical bank notes and coins, the rest is stored in the IT systems of commercial banks.
Your bank balance is simply a number linked to your account ID and stored in a computer. This allows for features like online banking where you are simply ask to send money from your account to another as a transaction (more on this later).
3) Banks have to be trusted
Banks prevent double spending by ensuring that you cannot spend the same money twice. If you have £80 and you spend £80 on Amazon, and then spend that same £80 on Ebay your payment to ebay is rejected because you have already spent your money on Amazon. Banks thus ensure that when you spend the money in your account it can only be spent once. If you spend it again it is either rejected or loaned to you as an overdraft.
If we could all stored electronic money there would be nothing stopping us from copy-pasting it to create new money. If you simply could simple store electronic money on your computer there would be a massive fallout from the counterfeiting of such money. Banks are thus the intermediaries that keep score on all our account balances.
This means that banks have to be trusted. They could simply edit their own balance or decide to issue themselves more money as and when they want it. This money printing devalues currencies and is often manipulated by central banks. In order for the system to stay standing we need a trusted central bank otherwise our economy would be destroyed in the same way as Zimbabwe and many others before it.
4) Banks make crazy profits from us - is there no other way?
This is the part where bitcoin comes into play. We have already covered that with traditional currencies you have to have a trusted intermediary in the form of a bank to allow transactions between two parties without those parties exchanging cash in person. This is known as a bank transfer for which the bank takes a hefty fee.
Bitcoin removes the bank and replaces it with another 3rd party - a distributed group of miners who are paid for keeping the system secure. Bitcoin allows a fully trustless system whereby you can send money from one side of the globe to the other in very short time frames without trusting any party in between.
5) Mining
Sending bitcoin from one person to another is a transaction. When a transaction is created it says:
I (wallet abcd1234) wish to pay wallet wxyz6789 X amount of bitcoin.
Rather than paying a bank to verify the transaction, it is broadcast to the network of miners (computers) supporting bitcoin. These miners have to put a significant amount of computing power towards verifying transactions to ensure no transaction can be faked or reversed, so they are paid a fee for their service.
As anyone can start 'mining' all the miners are competing to verify transactions. This keeps transaction costs down to an absolute minimum.
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You may point out that I have not explained the details around mining, only that they are required to keep the network secure, and paid for their services. If people then ask about where the bitcoins are created you can simply add that a limited supply of new bitcoins are given as a reward to miners for securing transactions.
There is no need to go into details about blocks and cryptography as frankly that goes deeper than most people need or understand. Unless they have a technical background it's better to just explain how bitcoins can be used by them.
This is a work in progress still, so if you want anything added or edited feel free to make suggestions in the comments. People are so used to currency that they take it for granted, bitcoin is a huge learning curve, and the media don't help as they tend to explain the mining process first in huge technical detail that makes bitcoin look ultra complex. One of the biggest things I try to get across to people is that in the same way that currencies are complicated, but cash is easy to use - the bitcoin protocol is complex, but using it is easy.
