There is no bank, no one in control of the Bitcoin transactions. How does that work?
When you want to open a bank account, you simply go to the bank, show your ID and other papers they need. They will create a unique account number for you and you are good to go.
People who need to send you money will use this account number to deposit money. You will use this number to pay your bills and withdraw cash. You can deposit to and withdraw from the account, other people can only deposit to your account.
In case you forget your account number, you can ask the bank for it. When you want to see transactions taken place on the account, you can use a statement of account.
Most of the above applies to Bitcoins too. There is no unique account number, but there is a unique wallet. Other people can deposit to your wallet, you can deposit to and withdraw from your wallet.
Who keeps score? No one and everyone.
When you start a Bitcoins wallet, you are also given a virtual notebook that holds all Bitcoins transactions that took place until now. Everyone who has a Bitcoin wallet, has this virtual notebook.
When someone wants to transfer Bitcoins from wallet A to wallet B, the transaction is broadcasted to the Bitcoinsystem. All computers that are part of this Bitcoinsystem will check their notebook. After checking, they will validate this transaction. The transaction will be performed after a certain number of validations. All notebooks will be updated with this transaction.
So everyone knows what is in my Bitcoin wallet? No, they can, if they want, check transactions for a certain wallet address. However, there is no identity tied to the wallet.
The Bitcoinsystem consists of a few servers that are always present, and wallet owners who are online. This means it is not one system that is maintained in a central place. It is a system that changes constantly.
Next one will be about the wallet. What is a wallet?