To understand what bitcoin is, you don’t need to know anything about math or computers. You only need to be familiar with money.
Money is a form of communication. When you are compensated for your work, you don’t get food, transportation, or lawn service. You get an unforgeable message, proof that you are owed. The medium isn’t important, the message may be printed on paper bills, or be sent by computer to update a bank account balance. The message serves a single purpose: you can present it to anyone who accepts this proof, and get back useful things that you are owed. The group of people who accept this kind of proof, we’ll call the 'consensus'. Without consensus, money is worthless. You can’t do anything with these messages except exchange them for something else.
The media often refers to bitcoin as a "digital" or "virtual" currency. This is wrong, computers are not what make bitcoin different. Most of the world’s payments have been digital for decades. Given how badly the media describes bitcoin, it’s no surprise that people don’t understand what it is.
Bitcoin is an idea: a "neutral" currency. In a "neutral" currency, everyone is treated equally - individuals, companies, governments, rich and poor. No special authority or trust is granted to anyone.
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No one can create money out of thin air
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No one can seize money that doesn’t belong to them
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No one can prevent others from spending their money as they see fit
Anyone can accept bitcoin as payment, and no one is forced to accept it.
Bitcoin (the technology) is a mechanism to reach consensus and communicate proof. It is a computer program, but you could imagine the same thing being done manually (and slowly) with paper and pencil.
Each participant saves the entire world history of messages. He opens a communication channel with other participants, and sends them copies of any history they might have missed. He follows a set of rules that decides whether each message is valid proof, throwing away any that don’t comply. Using all the valid messages, he can come up with a list of all the participants, and how much each of them is currently owed.
By following the same rules, participants come up with exactly the same list, forming a consensus and a common currency. If someone decides to play by his own rules, the consensus will reject his messages and they’ll become worthless.
Any participant or group can decide to break from the bitcoin consensus by following different rules, it doesn’t hurt bitcoin at all. There’s no such thing as "counterfeit" bitcoin. It has just enough rules to create money, and no more.
Your everyday national currency (eg the US dollar) is not neutral.
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You are forced to accept dollars. You must pay taxes, and you must pay them in dollars. You can’t do that without accepting dollars at some point.
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The dollar does not treat all participants equally. It grants powers to a small minority. The central bank prints money at no cost (seignorage), government agencies and banks can (and do) seize money from other participants. You have to trust these entities not to abuse their power, but inevitably they do abuse it.
- Price volatility
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There is a lot of uncertainty about how large bitcoin’s consensus will get. Bitcoin is voluntary and relatively new. It started in 2009 with no consensus at all, and is spreading organically (roughly 1 million people as of 2014). Eventually everyone in the world will decide if they want to accept bitcoin or not, and then there will be no more uncertainty.
- Cheap transactions
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This is one of the most-touted aspects, but it is a consequence of technology, not consensus. Today, every modern home computer can single-handedly keep up with the entire world’s bitcoin transactions. That’s very cheap, but it’s possible for bitcoin’s growth to outpace the capacity of computers and networks. In that case, transactions have to get more expensive (to discourage smaller transactions from clogging up the works), or bitcoin has to lose some of its neutrality (if your machine can’t keep up, you’ll have to trust someone else to do it. That gives them the power to lie to you).
- Anonymity
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Another aspect that is due to technology, not consensus. Bitcoin is currently not anonymous, it’s pseudonymous. There are only account numbers, no real names. It’s common to use a new account for every payment you receive. That makes it very difficult for authorities to determine which accounts belong to whom, but not impossible. True anonymity is theoretically possible, but so far it is too resource intensive, this is an area of active research.
As long as there are people who believe in neutral rules, neutral money will exist. All that’s needed are a set of rules, and the ability to communicate and process messages (an average computer and internet connection will do nicely). The genie is out of the bottle, and no amount of propaganda or regulation can put it back in.