How do off chain transactions work? Some wallets (such as inputs.io) claim to provide off-chain transactions. Note: I have not used that wallet yet.

Say if I have a Bitcoin address A in such a wallet provider and send 1 Bitcoin to address B from the same wallet provider, what will show on the public blockchain?


Will there be 1 Bitcoin in B on the public blockchain?
If yes to 1, will that Bitcoin show as coming from A?
There will have to be a public record of a transfer into address B before any transfers can come from address B.  Internally the provider may move the funds from one address to the other "off-chain" for purposes of display but until some BTC moves into address B publicly it can't be spent from B.

Most likely when you attempt to send BTC externally from address B, that amount will either be:

1) transferred publicly from A to B and subsequently from B onward (incurring 2 transactions and twice the confirmation delay)

2) transferred to a provider address from address A and be internally deducted from address B inside the provider system and actually transferred from some other provider address

3) publicly paid directly from address A but shown as deducted from B in the provider system  

Off-chain transfers are one way to add additional anonymity to Bitcoin transactions but at some point the source address has to be publicly credited with BTC for the exchange to be valid in the block chain.