Your private key is an encryption/decryption key known only to you. With this key you are able to do everything you can with a bitcoin address (spend, sign etc).

Using bitaddress.org:

Paper wallet - WITHOUT BIP38:
You are only given private keys that you must store

Paper wallet - With BIP38:
Your private keys are given encrypted and you must store the encrypted private key and the passphrase

Brain Wallet:
Your private keys are determined by the passphrase alone so you only need to store the passphrase (passphrase needs to be very strong to avoid brute force attacks)