What BitCoin is : Bitcoin is an Internet currency. Unlike fiat currency, no central authority can print money at will. The total number of Bitcoin in existence at any time is known to the Bitcoin community and controlled by a known algorithm. Because it is based on military-grade cryptography, Bitcoin is impossible to counterfeit. One Bitcoin (or BTC) can be divided into as small of pieces as you want (down to 0.00000001 BTC) 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.
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.