How will this answer change when Casper is deployed?
In general, a "confirmation" of a transaction receipt is an economic assurance or "economic proof" that it is correct.
Casper's "confirmation" strategy is to provide transaction receipts that include the bets that validators have placed on the correctness of receipts (bets on state), and on the guarantee that the transaction will be executed (bets on blocks). A client will choose the receipts that have the most backing from the bonded validators. Once a client sees a finalized receipt, they have reached "maximum security", and they stop choosing between receipts!