What is a Bitcoin Address?
A Bitcoin address is similar to the account numbers in a bank. You can send money bitcoins to any address, just like you can send cash to any bank account if you know its number, and you can only spend bitcoins from an address that you control. You control your Bitcoin address by a secret key, called the private key. Unlike an account in a bank though, you can easily have as many Bitcoin addresses as you wish.
What is a Bitcoin Wallet?
A Bitcoin wallet is a computer program that contains the private keys of your Bitcoin addresses and makes transfers in and out of those addresses easy and secure. A wallet also shows you the balance of all bitcoins that exist in the Bitcoin addresses that it contains. You should protect your wallet with a strong password and make sure nobody else has access to it. This is very important and it has to be repeated: You should never give out the private key associated with a Bitcoin address! If you let someone else get hold of a private key, they will be allowed to spend your Bitcoins from your bitcoin address. It’s a little scary to realize this, but once you do it’s no big deal; just protect the keys by encrypting them, something that your wallet makes convenient.
See choosing a wallet.
What’s the deal with cryptography and security?
You might read a lot of descriptions about Bitcoin that mention cryptography, hash functions, public and private keys and all sorts of very geeky sounding terms. For starters the only thing you really need to understand is that Bitcoin works and your bitcoins are secure because of complicated mathematics. This complicated mathematics allows you to conduct business and exchange money with people on the Internet who you don’t know and who you don’t necessarily trust. And that is a really cool thing!