Safest Bitcoin wallet
The answer from AntonAnsgar 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 reover your bitcoin in case you lose your offline wallet or password (computer failure, fire, etc.). I recommend storing it in a bank safe if it holds any substantial value.
I also wanted to add that a new, more handy way of storing Bitcoin safely is coming soon, it's called Hardware wallets. Hardware wallets are a small piece of hardware which store your private key securely and only allow signing transaction, usually with a confirmation prompt 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 verify the transaction output addresses before letting the device sign it, and the PIN protects the device from theft. There are multiple different implementation, but one that appears to me as the most open and promising is Trezor which have said they'll start shipping their initial pre-order devices this month or the next.
Offline wallets are probably still the best way to store large amounts securely, and the upcoming hardware wallets devices are going to be a life changer for day-to-day bitcoin transactions making them even more secure to theft than chip-protected debit and credit cards.
NB: I'm in no way affiliated with Trezor; I haven't mentioned others as only Trezor appears to be almost ready to ship and easy enough for most people to use, plus they promised to release the code as open-source before shipping the devices.
