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.
The network is always trying to extend the longest chain, and eventually, some chain will win. The deeper the transaction is in the chain, the higher the chance it will win because the network tries very hard not to duplicate efforts. (Because miners want their coinbase transactions to win, they all try very hard to extend the chain most likely to win, which makes sure it does in fact win.)
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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