What is "the Blockchain"?
Technical explanation: The Blockchain is a ditributed database designed to maintain a continuously growing list of records, called 'blocks'. Each block contains a timestamp and a link to a previous block. The Blockchain is a series of transactions (or chain of blocks), each with a recorded ledger of validated electronic promises to pay (eIOU). All network (or BitCoin) users have a copy of the Blockchain, which represents the agreed upon version of the 'truth'.
And here's an explanation, based on an analogy of everyday's life. This is how the blockchain would work in an analog world: Let's suppose you go to a grocery store in a huge shopping mall. You only have 1 monetary unit (be that a silver coin, a $50 note or whatever). You choose the products you want to purchase and go to the checkout. You send an eMail to the cashier stating that you promise to pay the amount of 1 monetary unit (MU) for the products that you are purchasing, and you BCC everyone in the mall. This eMail is your IOU ("I owe you"), your promise to pay. It is, by the way, the exact same thing that you give to the bank and get a freshly generated amount of money as a loan in return. In the Blockchain, you are your own bank.
Back to the scene in the store: Witnesses who happen to be in the area received the copy of your eMail and are eager to confirm this transaction because they get a little share, so they immediately confirm that you have what you say you have, and that you will pay the cashier the amount you say you will. From this moment on, this becomes a confirmed transaction. The other customers in the mall will then get an eMail stating that this transaction is valid, and they will, for their part, put everyone else in copy as well.
After close of business, the cashier goes to his bank and the bank credits $50 to his account, since also the bank got an eMail confirming that the transaction is valid, and that you will keep your promise.
But why should you? You still have your monetary unit in your wallet. So let's next go to a different place and spent it, right? You choose a product again, you head to the checkout, you make again the promise to pay, but this time it gets rejected. Why? Because that merchant also received an eMail and knows that you already used the monetary unit you had to pay for your purchases in the grocery store, and that you have no money left. Everybody else is also in the loop and can confirm. The transaction will never be validated.
Now let's say you find a cashier that doesn't like his boss and doesn't care whether you have money or not. He maliciously accepts your promise to pay for the products you purchase, because it's only his boss's money and his intent is to harm his boss by giving away his products for free. What will happen in such a case is that the other customers in the mall will blow the whistle and expose that false transaction. Even if you and the clerk want to convince the rest of the people that it actually is valid, you'd have to convince 51% of all the people, which economically makes zero sense. It would be much easier to work for an hour and get the money to pay for the goods. And as for the cashier, he will have to find other ways to harm his boss's business.
Every transaction that is made and confirmed gets added to the confirmed transactions (blocks) from the past. The result is a chain of blocks. A Blockchain. Like a public book keeping ledger. It can't be altered, it can't be changed, and every single user gets an exact copy of that book that matches all other copies. If someone shows up with an altered or outdated copy that doesn't match the other copies, this copy is automatically replaced by one that matches all other copies.
All this is done not by corrupt politicians or greedy banksters or by their unmotivated employees or slave labourers in some industrial complex in China, but by the machine.