In a nutshell 'Bitcoin' is a transaction ledger of units. The actual ledger data is known as the 'blockchain' and it is kept virtually, peer-to-peer via the internet and not owned by anyone but stored by each individual who keeps it alive. It goes by a strict set of rules (the core of which is zero fees) and every transaction goes into micro-escrow to validate with the entire blockchain that the transaction meets the rules before it can proceed. If anyone tries to change the system they are ignored. The code for the system is open source and is constantly being improved by the Bitcoin foundation who make zero money from the system with all code published publicly and code signed. The units of this ledger are called Bitcoin, but can be subdivided down five or more orders of magnitude to a 'satoshi'. Each subdivision of a Bitcoin is owned by whatever 'wallet' address they are assigned to. The owner of that wallet is whoever keeps its 'private key'.
Bitcoin is identical to the concept of downloading files with torrent magnets. The files are not hosted by anyone and the system is started and kept alive by individuals who benefit from the system. The files are checksummed and if someone tries to change any file, that change is ignored by everyone else.
Now about the integrity of Bitcoin. And this is where it gets interesting as it highlights not currency but something far more important. Something political, or social - nay, anthropological. It shows a way forward for the human species. It showcases the corruptibility of people but also shows how to avoid this corruption and reward us into all working for a common good.
When humans aggregated into civilisations, we were ruled over by monarchies. Businesses independent from the monarchy existed, but only where they would benefit the monarchy. When individual endeavours became large or threatening enough they were dealt with by the monarchy through excessive taxes, acquisition and elimination, or if they were too powerful they were 'absorbed' into the monarchy as barons. Not bloodline but part of the 'family'. Only those in the monarchy had god given rights and the average person had no individual rights within the context of their society. Moving forward through the dark ages, society's thinking advanced and by the 1500's people started wanting something for themselves. The loudest voices for change were Jewish merchants who were brilliant at setting up independent businesses that thrived. As soon as these businesses were substantial enough, they were then taxed, absorbed or destroyed by the ruling monarchies as they saw fit. The cry against this injustice spoke out for the notion of individual rights. Liberalism was born. The figurehead for this movement was John Locke and his "An Agreement for the People" document became the basis and the core tenets for all democratic constitutions and governments through to today. (as a side note this is where the evil of Google is being overlooked - Google's electronic invasion of privacy has effectively undermined key pillars of our liberalistic constitutions - the right to individual privacy and the right of free speech - but thats another story).
Over the next two hundred years the Age of Enlightenment blossomed. The Jewish merchants rode this wave of enlightenment and set up 'Banks' to give them and all people, the monetary freedom from monarchistic control and manipulation. Which it did.
But what happens when you give people monetary power? Just as with the monarchies, monetary power creates corruption and now the most controlling and manipulating force in the world are banks and monetary organisations. Monarchies have withered and governments are now the puppets of banks and money. Even worse, the growing belligerence of banks is now being felt by each individual. We are being stolen-from by them at every step. We are losing our homes because of their greed. And apart from cash, we can't pay anyone without banks taking a cut. And ironically now in a technological age where one can create multimillion dollar products (even something as sophisticated and capital intensive as a movie) which can be sent electronically to anyone for free - the seller cannot accept payment for that product without it going through an opaque labyrinth of banks and organisations taking their cut. Even buying a book on Amazon delivered here to Australia, means that 9.5% goes to a bank somewhere (3% merchant fee, 1.5% Int transaction fee, 5% inflated currency conversion rate). So we need a technological solution to money transactions. One that avoids the corruptibility of people and the fraudulent organisations that steal from our every non-cash transaction.
Introducing peer-to-peer networks. One side effect a of peer-to-peer system is that they can even get the lowest form of criminal today, the internet's petty thieves - file downloaders, to contribute to a system by uploading to all, in order to get what they want for free. The rules of the peer-to-peer system are not broken as the system benefits the individual too much. Public torrents distort this with 'leechers' contributing very little, and people with unrestricted internet contributing disproportionately. But add one simple rule to the system - that one must upload the exact amount or more that what they download. Then the system equalises completely and distortions of greed are eliminated and the members of the system still get what they want for free and they still contribute. These same low lives will never think to compromise the system because it benefits them too much, to them its a win-win, even though its for the greater good of the system.
And thats how Bitcoin works. Its trial by fire was its use by Silkroad by criminals selling drugs, stolen good, arms and even hit-man services. But not one of these criminal elements undermined the system, because it benefitted them too much. It would be like criminal individuals trying to destroy the system of the US dollar. They wouldn't want to, because cash in US dollars is their prime method of payment. Less than savoury examples surely, but important because they show that even criminal elements can be made to contribute to these systems.
And thats how we as a species can move forward and up. Remove the points where corrupt individuals can exert power, replacing them with a cryptographic system with given, transparent rules that we understand and agree to and where each individual benefits. Then it is part of our very nature to support such a system. Even the lowest amongst us.
The recent incident with the QLD power grid going surplus from home generated electricity is also an example of a peer-to-peer system at work and a compelling possibility for the future. Now imagine if each house had to install solar panels that covered at least all of its own power requirements, with the excess being given back for free to everyone else (or sold to businesses). Then our power problems are solved. There are some issues such as storage and cabling, but in the scheme of things these are relatively minor. How we benefit is that the corrupt power holders (electricity suppliers who currently charge a 400% markup on the actual cost of electricity) are removed and our power is then free and non-polluting for all.
How many other areas can cryptographic peer-to-peer networks be applied to remove corruptible people and improve how we do thing? Voting and Government policies? Traffic management, product and pharmaceutical development, gene research and scientific modelling?
And that is why Bitcoin is so interesting. It is not only because of its use as a currency, its because it is a beta test for an underlying technology and overlying human-psychological system which can solve massive human problems, to the benefit of us all.
Wikipedia on Liberalism Our rights as an individual haven't been around that long
Bitcoin White Paper Just the first page will give you an idea of what Bitcoin is aiming for