I've heard of people losing bitcoin because they threw out a hard drive. And I've heard that you can control bitcoin with just a brain wallet (a memorized passphrase). Do you need just a passphrase, or do you also need a wallet.dat file (if you're not using an online wallet)? If someone could provide a simple explanation or point me to one I'd appreciate it.
A wallet holds one or more private keys. It does not hold bitcoins.
Holding a private key allows you to send bitcoins from an address.
A brain wallet is simply the case where you memorize a private key. In this case, your brain holds a private key so it is a wallet by definition -- a "brain" wallet. However, in this case you memorize a passphrase that is used to generate the private key and not the actual key.
People lose bitcoins when they lose a wallet file because they haven't backed up the wallet file or the private keys that it contains.
Please note that the passphrase used to unlock a wallet is simply the passphrase used to encrypt and decrypt the wallet file. It is not the same as a private key or a brain wallet pass phrase.
Good answer, thanks. So, as I understand it, you only need the private key to send bitcoin from an address, whether that private key is stored on paper, in a wallet file, or in your brain. I was just reading the bitcoin wiki and it said that the address can be generated from the private key, so apparently you don't even need to store the address somewhere if you have the private key.