What is Bitcoin?

It's a common question:
"What is Bitcoin?"
Well, it's not exactly a simple answer.
Bitcoin is a peer-to-peer crypto-currency - this means it's a completely online currency that is protected by strong cryptography.  Created in 2009 by a person or group using the alias Satoshi Nakamoto, is has no government or financial institution backing it. It's value is purely derived from the value others place on it - not far removed from our current fiat currency. ;)
And everything is verified by a huge network of users, completely anonymously.
So, as a user you can create a 'wallet', which allows you to set up a specific address bitcoin can be sent to as well as send from.  This is essentially your 'bank', and there are versions online as well as for your computer.
Users who download a bitcoin 'client' also sync with all the data (called 'blockchain') in the world.
Here's how it works:
I want to buy a pizza from you, and you ask for 0.014 bitcoin for it.  You give me your 'address' (or a QR code to scan), and I use my wallet to send you that 0.014.
This transaction is automatically recorded on everyone elses bitcoin client as well, almost eliminating cheating the system, because if I tried to send that same 0.014 to someone else, thousands or millions of copies will forbid it since those bitcoin have already been sent!
You receive 0.014 bitcoin and can now 'spend' it as you wish.  I have 0.014 less.
And everyone involved has a copy of it, with no transaction fees or middlemen.
Importantly, it's completely anonymous!  Only the addresses involved are recorded, no personal data!
Bitcoin mining
Now, not every bitcoin was created all at once, and there's a limit to how many will ever be produced. The specifics are beyond this article, but the creation of 'new' bitcoin is called "mining", and anyone can do it.
Complex calculations are done verifying all transactions in a shared, public database, and when a new 'chain' is unlocked successfully, an award of 25 bitcoin is given.
Unforunately this has now entered industrial-level mining, as the price of bitcoin has skyrocketed to upwards of $1000 US at the time of this writing, so the complexity of the calculations required have also risen dramatically.  You could try to solo mine bitcoin, but it could take months with no guarantee that you would be awarded the newest 25 bitcoin!  The average user's only hope is to join a mining 'pool' and work together, increasing success, but rewarded with small fractions of bitcoin.