The answer from Anton A. is pretty good; I'm adding this as additional information.
Offline wallets are stored on a computer that never connects to the network so they can only be hacked by being physically present. Additional security can be added by encrypting the Bitcoin wallet or the computer as a whole (be aware on the later that the encrypted computer is only secure once it has been completely shut down for several minutes).
The way it works with Armory is that you have an online copy with the public key where you can see your balance. To create a transaction you do it on the online wallet, copy it to a USB stick, connect it to the offline computer and sign it there, then once you load the signed transaction on the online wallet it gets sent to the Bitcoin network for inclusion in blocks.
The paper backup is needed to recover your Bitcoin in case you lose your offline wallet or password (failure, fire, etc.). I recommend storing it in a bank safe if it holds any substantial value.
Another type of wallet that provides a new, handier way of storing Bitcoin safely is called a Hardware wallet. Hardware wallets are a small piece of hardware which store your private key securely and only allow signing transaction, in some case with an address confirmation display and PIN. Just like offline wallets, hardware wallets allow you to save a paper copy of your key in case you lose or break the device.
These devices are safe to use even on a compromised computer provided that you can verify the transaction output addresses before letting the device sign it, and a PIN may protect the device from theft. There are multiple different implementations with varying levels of security. Some of these wallets have additional features such as providing 2-factor authentication.
Offline wallets are probably still the best way to store large amounts securely, accompanied with either a mobile or hardware wallet for day-to-day Bitcoin transactions.