Bitcoin confirmations represent the number of blocks in the block chain that have been accepted by the network since the block that includes the transaction.
In simpler terms it represents the difficulty of a double spend attack. With zero confirmations no proof of work has been done, so you can't tell if anyone considers the transaction valid. Even with a single transaction it is possible for an attacker to pre compute a single block.
An attacker must match the power of the entire bitcoin network to keep up with block creation so as time goes by it becomes increasingly difficult to forge a transaction.
If you don't grasp the basic concept, imagine for a second that a network outage split the bitcoin network in half. I could send one transaction giving 30 bitcoins to Abel to one half and one transaction giving those same 30 bitcoins to Fred to the other. Each half would accept that transaction and until the two halves reconnected, you wouldn't know which transaction would be honored tomorrow.
Confirmations are simply blocks that have been generated after the block that contains your transaction. Because there is no central authority that can be consulted to be sure a transaction will be committed, recipients use the number of confirmations as a way of protecting against double-spend attacks.
Effectively, the more confirmations, the higher the likelihood that a transaction will remain forever in the public hash chain rather than a conflicting transaction if there was one. At 6 confirmations, it is perhaps one in a billion that a transaction won't be permanent -- and that's if the sender is attempting a double spend attack.
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