Your note asks something different than the title: how to apologize in a professional way that "returns to normal" quickly. This has been a pet goal of mine for awhile, I hate when people dump on me for a fluke. And, for the love of god, people (especially superiors) LOVE explaining how to do things even if it's obvious. Here's what I got:
First: don't make the mistake -- think ahead, ask, etc. If it's a time thing, ask for more. If it's a hard deadline, let affected people know as soon as it goes wrong.
Second: downplay it if it's small (NOTE: these will only work if your job allows for small mistakes). Seriously, why do you need to be someone's emotional dumping ground because you forgot to add an attachment, or whatever? People can take a cue from you on how big of a deal this is, and you can make an emotional avalanche out of a snowflake. "You didn't send the doc". "You're right, sending now." Done, no apology necessary. If it's a teacher or superior, it's an opportunity to remind them of all your successes -- "Ach, there were a thousand different things I exceeded expectations on and of course I slip on the most visible one. Fixing now." Make the mistake feel as if anyone would do it in your shoes. "Boss, I was swamped by the reports yesterday, I've amended it for your review. I'll be swamped with reports again today." If it's a peer, they need to know that they can trust you. Remind them that the job is taxing and they've made mistakes. "Ach, you're right, I missed it. I guess were even now from that deadline snafu of yours." Don't be catty, snarky, biting, or competitive, just have an edge. Peers won't see you as inferior if you call them out on their mistakes too, because that's essentially what they're doing (don't do this with bosses though).
Third: if the mistake is not small, portion your response according to the size of the mistake. The rule is: the size of the mistake is equal to the impact it has on them. "Boss, apologies for not sending the report on time. It's attached here." Don't elaborate their position for them, that's their job, and letting them vent is your price to pay for making a mistake. (Plus you could be wrong on their position). NOTE: if there is a legitimate reason for your mistake, this is your first line of conversation. "Some asshole totalled my car this morning, I'm lucky to be alive." "Is the report finished?" "No, my cell phone was in the car, I had my friend drop me off because I needed to tell you."
Fourth: Fix it. Right now. It only gets worse as time passes. If they keep talking, then you can point out that it's already fixed.
FINALLY: you can't make more mistakes than everyone else and expect these tips to work. You'll get branded as incompetent. I make mistakes, sure, everybody does. But I effing can't stand incompetence. I'd fire it, and chances are, your bosses would too.
(The replies here are mostly about triaging emotions from a small mistake, but good luck trying this if you busted someone's $1mm deal.)
ONE IMPORTANT NOTE: Some mistakes are circumstantial, i.e. a total fluke, or basically any human under in those circumstances will have made a similar mistake. A good manager will look for ways to improve, seeing mistakes as a byproduct of a process that needs improvement. A bad manager... will look for someone to unload on.
*TL;DR -- Competence is the best way to silence superiors. If the mistake is small, paint it as normal/smaller than they are; don't be a punching bag, don't apologize a lot; portion your response according to the damage they feel; and NONE OF THESE WORK if you make a *giant mistake.