What is wallet?

To use Bitcoin, you need a wallet. A wallet is essentially an address you use to receive BTC. Whenever you acquire a new wallet, a private and public key is generated and given to you. The public key can be freely shared and is what people use to send BTC to each other. This public key is stored with the amount in the currency's transaction history, known as the block chain. By looking at the block chain, you can determine how much BTC an account has by adding up how much was sent and received to that account. However, the private key should be kept secret as whoever knows the private key claims ownership of all its associated funds (in fact, by default the local wallet hides the private key from you so you don't accidently reveal it). Whenever you back up a wallet, you actually aren't saving how many coins you have, but the private key. This way, you can't spend coins then restore from a backup to regain them. Instead, the private key is used to just claim ownership of all funds associated with its public key pair in the block chain.