The A in ACID. It means that an operation should either execute completely or have no effect at all. It should never be executed partially.
F.ex, if a bank's DataBase system is shuffling money between accounts, subtracting money from the source account and adding it to the destination account should be atomic. If it fails, it should fail completely. You don't want to end up with money vanishing by being subtracted from one account, but not added to any other. Nor do you want to end up with money magically sprining into existance by being added to one account but not having been taken away from any other.