Confirmations correspond to the number of blocks that are dependent on a given transaction. This includes the block containing that transaction and all blocks after it in the blockchain. The more confirmations a transaction has, the more likely is it that it will remain in the blockchain permanently.
On Ethereum specifically, "12 confirmations; however, exchanges and entities handling very large amounts of Ether frequently are still encouraged to run two different Ethereum implementations and only accept transactions that have been confirmed by both for maximum security (e.g. Go & C++)."