Sites or users using the Bitcoin system are required to use a global database called the blockchain. The blockchain is a record of all transactions that have taken place in the Bitcoin network. It also keeps track of new bitcoins as they are generated. With these two facts, the blockchain is able to keep track of who has how much money at all times.
To generate a bitcoin, a miner must solve a math problem. However, the difficulty of the math problem depends on how many people are buying bitcoin at the moment. Because of how complicated the math problems usually are, they must be calculated with very powerful processors. These processors can be found in CPUs, graphics cards, or specialized machines called ASICs. The process of generating the bitcoins is called mining. People who use these machines to mine bitcoins are called miners. Miners either compete with one another or work together in groups to solve a mathematical puzzle. The first miner or group of miners to solve the particular puzzle are rewarded with new bitcoins.
The puzzle is determined by the transactions being sent at the time and the previous puzzle solution. This means the solution to one puzzle is always different from the puzzles before. Attempting to change an earlier transaction, maybe to fake bitcoins being sent or change the number of someone's bitcoins, requires solving that puzzle again, which takes a lot of work, and also requires solving each of the following puzzles, which takes even more work. This means a bitcoin cheater needs to outpace all the other bitcoin miners to change the bitcoin history. This makes the bitcoin blockchain very safe to use.
A popular image associated with Bitcoin is a QR code. QR codes are a group of black and white boxes that are similar to barcodes. Barcodes have one dimension of information, while QR codes have two (horizontal and vertical). Barcodes are a row of lines, and QR codes are a grid of squares. Bitcoin uses QR codes because they can store a lot of information in a small space, and a camera such as a smartphone can read them. The two QR codes on the Bitcoin note are the public and private addresses, and can be scanned with a number of online tools.
Everyone in the Bitcoin network is considered a peer, and all addresses are created equal. All transactions can take place solely from peer to peer, but a number of sites exist to make these transactions simpler. These sites are called exchanges. Exchanges provide tools for dealing in Bitcoin. Some allow the purchase of Bitcoin from external accounts, and others allow trading with other cryptography-based currencies like Bitcoin. Most exchanges also provide a basic "wallet" service.
Wallets provide a handy way to keep track of all of a user's public and private addresses. Because addresses are pseudo-anonymous, anyone can have as many addresses as they want. This is useful for dealing with multiple people, but it can get complicated to manage multiple accounts. A wallet holds all of this information in a convenient place, just like a real wallet would. A backup of a wallet prevents "losing" the bitcoins.