Bitcoin confirmations are the number of blocks in the blockchain that have been accepted by the network since the block that includes the transaction. Literally, 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.
After a transaction is broadcast to the Bitcoin network, it may be included in a block that is published to the network. When that happens it is said that the transaction has been mined at a depth of 1 block. With each subsequent block that is found, the number of blocks deep is increased by one. To be secure against double spending, a transaction should not be considered as confirmed until it is a certain number of blocks deep.