I am an utter newbie when it comes to BitCoin. I am consideirng using the Armory wallet. The thing I do not understand - if the wallet is offline (in the sense of my computer being switched off) what happens to payments made to that wallet? Do they just disappear into the ether or do I have the ability to recover them when I go back online?
Wallets don't store bitcoins. Wallets store private keys. Private keys are a bit like passwords. They allow you to create a digital signature that proves to the entire network that you control a particular bitcoin address.
All transactions are permanently stored in the ledger known as the blockchain. Every full node on the bitcoin network contains a complete copy of the blockchain.
When a transaction is created to send bitcoins to your address, the person who controls the bitcoins being sent creates a transaction with their wallet. Their wallet uses the private keys that it owns to sign the transaction and then broadcasts the transaction onto the bitcoin network.
Every node on the network verifies that the signature is legitimate and then eventually a miner adds the transaction to the blockchain. Once that happens, the only way those bitcoins can be transferred anywhere else is for a new transaction to be broadcast with a digital signature from the private key associated with your address (a private key from your wallet).
When you turn your computer back on (or bring it online), it will synchronize its copy of the blockchain with the peers that it connects to. When it sees the transaction in the blockchain that assigns bitcoins to an address that it knows about, it will update the balance that it displays to you, and allow you to create transactions to send that value elsewhere.
Armory has the ability to have it's functionality split between two computers as well. There is an "online" portion that runs on a computer that is allowed to synchronize the blockchain but that doesn't have any of the private keys at all, and a permanently offline portion that runs on a computer that is never connected to the internet at all and stores all the private keys. When using this functionality, you create an unsigned transaction with the online portion. Then you transfer that transaction to the offline computer (via removable USB storage) to be signed. Then you transfer the signed transaction back to the online computer (via removable USB storage) to be broadcast to the bitcoin network.
