Forget most things you've heard. People discover Bitcoin in a variety
 of ways, but usually pick up some sort of misconception like "Bitcoin 
gives free money to people with computers" or "in order to use Bitcoin I
 have to use a program that wastes electricity for nothing" along the 
way. Here is a good summary to help you understand Bitcoin in general, 
by focusing on what Bitcoin is and what problem it solves. These two 
things are not typically well explained on most websites, and it is 
difficult to appreciate just how effective a technology Bitcoin is until
 they are understood.
What Bitcoin is: An agreement amongst
 a community of people to use 21 million secure mathematical 
tokens--"bitcoins"--as money, like traditional African and Asian 
societies used the money cowry. Unlike the money cowry:
- there will never be more bitcoins
- they are impossible to counterfeit
- they can be divided into as small of pieces as you want
- and they can be transferred instantly across great distances via a digital connection such as the internet.
This
 is accomplished by the use of powerful cryptography many times stronger
 than that used by banks. Instead of simply being "sent" coins have to 
be cryptographically signed over from one entity to another, essentially
 putting a lock and key on each token so that bitcoins can be securely 
backed up in multiple places, and so that copying doesn't increase the 
amount you own.
Because bitcoins are given their value by the 
community, they don't need to be accepted by anyone else or backed by 
any authority to succeed. They are like a local currency except much, 
much more effective and local to the whole world. As an example of how 
effective the community is at "backing" the bitcoin: on April 4th 2011 
30,000 bitcoins were abruptly sold on the largest Bitcoin exchange, 
consuming nearly all "buy" offers on the order book and dropping the 
price by nearly 1/3. But within a couple of days, the price on the 
exchange had fully rebounded and bitcoins were again trading at good 
volumes, with large "buy" offers slowly replacing the ones consumed by 
the trades. The ability of such a small economy (there were only 5 
million out of the total 21 million bitcoins circulating then, or about 
3.75 million USD worth at then-current exchange rates) to absorb such a 
large sell-off without crashing shows that bitcoins were already working
 beautifully.
What problem Bitcoin solves: Mathematically, 
the specific implementation of the bitcoin protocol solves the problem 
of "how to do all of the above without trusting anyone". If that sounds 
amazing, it should! Normally a local currency has to trust all kinds of 
people for it to be able to work. So does a national currency. And in 
both cases, that trust is often abused. But with Bitcoin, there's no one
 person who can abuse the system. Nobody can print more money, nobody 
can re-use the coins simply by making a copy, and nobody can use anyone 
else's coins without having direct access to their keys. People who 
break its mathematical "rules" simply end up creating a whole different 
system incompatible with the first. As long as these rules are followed 
by someone, the only way Bitcoin can fail is for everyone to stop using 
it.
This marvelous quality of not having to trust anyone is 
achieved in two ways. First, through the use of cutting-edge 
cryptography. Cryptography ensures that only the owner of the bitcoins 
has the authority to spend them. The cryptography used in Bitcoin is so 
strong that all the world's online banking would be compromised before 
Bitcoin would be, and it can even be upgraded if that were to start to 
happen. It's like if each banknote in your pocket had a 100-digit 
combination lock on it that couldn't be removed without destroying the 
bill itself. Bitcoin is that secure.
But the second way of 
securing the system, called the blockchain, is where the real magic 
happens. The blockchain is a single, authoritative record of confirmed 
transactions which is stored on the peer to peer Bitcoin network. Even 
with top-notch digital encryption, if there was no central registry to 
show that certain bitcoins had already been "paid" to someone else, you 
could sign over the same coins to multiple people in what's called a 
double-spend attack, like writing cheques for more money than you have 
in your account. Normally this is prevented by a central authority, the 
bank, who keeps track of all the cheques you write and makes sure they 
don't exceed the amount of money you have.
Even so, most people won't 
accept a cheque from you unless they really trust you, and the bank has 
to spend a lot of money physically protecting those central records, 
whether they are kept in a physical or digital form. Not to mention, 
sometimes a bank employee can abuse their position of trust. And, in 
traditional banking, the bank itself doesn't have to follow the rules 
you do--it can lend out more money than it actually has.
The 
blockchain fixes all these problems by creating a single master registry
 of the already-cryptographically-secured bitcoin transfers, verifying 
them and locking them down in a highly competitive market called mining.
 In return for this critical role, the Bitcoin community rewards miners 
with a set amount of bitcoins per block, taken from the original limited
 quantity on a pre-agreed schedule. As that original amount gradually 
runs out, this reward will be replaced by fees paid to prioritise one 
transaction over another--again in a highly competitive market to ensure
 the lowest possible cost. The transactions are verified and locked in 
by the computational work of mining in a very special way so that no one
 else can change the official record of transactions without doing more 
computational work than the cumulative work of all miners across the 
whole network.
In conclusion: All this mathematical 
technology may be a bit of a mouthful, but what it means in practice is 
that Bitcoin works just like cash. Bitcoin transactions are 
intentionally irreversible--unlike credit cards or PayPal where 
chargebacks can invalidate a payment that has already been made. And 
there are no middlemen. Transactions are completed directly between the 
sender and the receiver via the peer to peer network.
Because of 
Bitcoin's intricate design, the network remains secure no matter where 
or how you process Bitcoin transactions. Which is incredible--no one 
else has ever tried to create a system that worked this way! All 
previous monetary systems have relied on trusting somebody, whether it 
was the king, town hall, the federal reserve, or banks. Bitcoin doesn't.
 It's guaranteed instead by the laws of mathematics, and that's why it 
has everyone from technologists to economists very excited. I'm sure you
 have lots more questions, so scan the index below to see if they've 
been asked before, then dive in! The so-called "canonical" threads 
linked from this index are considered newbie-friendly zones; outside of 
them you're welcome to try your own luck.
                                                                                                       - BitTalk.org