In baseball, the big mistakes are called errors. The result of an error is 60,000 people deriding you with boos and maybe throwing cups at you from the stands. The proper player's response, in this situation, is to never, ever repeat the offense. The consequence of ignoring the mob and the flying cups is basically losing your job.
In business, you won't be subjected to the cacophony of 60,000 disenchanted ticket payers. But when customers or co-workers barrage you with their disappointment, it might feel like that. Go ahead, you think, throw the cups. Face it: You've dropped the ball. Now what?
A professional apology should be relative to the crime. If you misspoke in conversation, correct yourself immediately by saying, "I'm sorry, I misspoke. I meant to say..." Not so big a deal. Move on. If you lost the deposit on the way to the bank, crashed the company car or a building you're responsible for was burned to the ground, then the apology might need to be more formal. You, your leaders and the media can decide together whether to hold a press conference.
Most mistakes fall somewhere in between. For your garden variety mess-ups, try this: Own it: "Here is what happened." Regret it: "I am sorry for any damage/injury/losses that resulted from my mistake." Repair it: "What can I do to make things right?" Then never let it happen again.
A researcher who studied boys and girls in gym classes across America discovered something interesting. Girls in the study apologized readily and casually when they make a mistake. Boys rarely apologized. The interesting part is that the girls never seemed very sorry; the boys felt awful for days.
So it seems your choices are to apologize and feel fine, or resist making the apology and feel lousy for days. Sometimes the best way through a tough situation is straight through it. Own it, regret it (for real), repair it and never, ever repeat it. Saying you're sorry gets easier with practice, and if we're honest, we all have opportunities to practice at least once or twice a week.