What is Bitcoin?
Bitcoin is money no one can take without your permission. It cannot be inflated away or confiscated, because no one person, company, or government controls it.
Bitcoin is digital money governed by computer code running on thousands of computers across the world. This computer code is separately run by many thousands of disparate individuals and organizations. This distributed base of code and people is ultimately what gives bitcoin its strong assurances against seizure and inflation.
Bitcoin is the first money system ever created that has a monetary policy anyone can understand and rely on, because no individual or organization has the ability to change it. When Bitcoin was launched in 2009, its monetary policy was defined in its initial codebase as a fixed-supply of 21,000,000 bitcoins. Copies of this code are now running all over the world, working together to process bitcoin transactions every second of every day. Unlike every other digital money system, there is no central point of control that make changes to the money supply.
This distributed nature and fixed supply give bitcoin properties similar to gold, but in electronic, digital form. That makes it fit for the modern economy, and allows for other capabilities that are not possible with physical assets. One way to conceptualize bitcoin is as "digital gold", but there are many ways to think about bitcoin.
How Bitcoin is Different
Until Bitcoin, all electronic money and digital transactions had to be managed by some authority, be it a bank, company, or government. Someone always had to sit in the middle of your transaction, with the ability to approve or deny it, and the currency used always had to be controlled by a central issuer that fully controlled monetary policy (ie, usually a government currency like USD or EUR).
This is not the case with bitcoin. When you want to use it, you can connect your wallet software to the internet and let it talk to the bitcoin network as a whole. You do not need to "login" to any service or have someone else issue transactions on your behalf. Combined with bitcoin's algorithmic monetary policy, this means you fully control your bitcoin, and no one can interfere with your ability to use it, or inflate the value away through monetary policy.
Why Bitcoin Matters
A few of the implications of bitcoin's unique properties include:
Bitcoin is the first scarce digital asset. Until bitcoin, anything digital could be easily copied. Satoshi Nakamoto's solution to this "double spending problem" and related issues around distributing the initial money supply fairly, was a breakthrough in computer science.
Scarcity is a key property of money. Until bitcoin, money native to the internet was not possible. With digital scarcity, native digital money can exist. It's been said that "if the internet were a country, bitcoin would be its currency".
Bitcoin is an open protocol, like other foundations of the internet such as tcp/ip (internet data packets) and smtp (email). Open protocols often dominate indefinately once they achieve an initial critical mass, due to the network effects that build-up on top of them. Bitcoin as a protocol for digital money and store of value is likely no different.
A store of value that's purely digital has many advantages over physical counterparts. Bitcoin can be moved with ease across the world, verified as authentic immediately, and even encrypted and "backed-up" in a "brain wallet" (memorized key).
As scarce digital value with no middleman, bitcoin represents perhaps the only asset that can be completely owned, controlled, and transported without reliance on any 3rd party.
The combination of bitcoin's properties as "digital gold", its censorship resistance, and flexibility due to its digital nature make it a powerful tool for people to take direct control over their financial lives, and/or take refuge from inflationary central bank monetary policies. This is why some describe bitcoin as "a peaceful revolution".
Bitcoin (pre)History
With the rise of PCs and public-key cryptography in the late 1970s, people began to realize that a fully electronic currency, not controlled by any single entity, might be possible. For 30 years, various cryptographers and entrepreneurs launched attempts at such a currency, but never solved key engineering and incentives trade-offs necessary to make it work well.
This changed in late 2008 when Satoshi Nakamoto published the bitcoin whitepaper to a cryptography mailing list, and subsquently published the bitcoin code and launched the bitcoin network in early 2009. Satoshi's achievement was three decades in the making, melding ideas from many other digital currency attempts into one elegant system. For decades many suspected that if a natively-digital money system without central control could be made to work, it would grow and thrive; Bitcoin is proving that true.