I'm not even going to ask if you have ever made a mistake, because I already know the answer. Everyone makes mistakes, but how you handle it makes all the difference.
Whether it is a big mistake or a small one, you probably should admit it to your boss. I say "probably" because sometimes, depending on the mistake, you can "fix" it and nobody really needs to know about it and it wouldn't be good to tell your boss about every little thing you did wrong. For instance if you are organizing a meeting and send the meeting request to the wrong person, you can easily fix it by apologizing to whoever you invited by mistake, cancel that invitation and then invite the correct person. Other times you really do need to tell your boss because it might have repercussions and it is best to admit it up front. You can determine whether you need to tell your boss or not, but you absolutely should admit it to yourself.
Learn from it
Once you have admitted the error you need to learn from it and do whatever you can not to do it again. I think sending emails to the wrong person is an easy mistake to make because it is so easy and quick to press Send, but not so easy to take it back (and the Recall function really does not work on emails to external people and is only hit and miss on internal emails). Depending on how often you make the mistake and how serious the consequences could be, will determine what kind of measures you need to take.
One way you can avoid sending an email to the wrong person is to turn off the automatic email memory function, then you will have to enter each person's email address each time and that will make it more difficult to make that mistake. You can also ask a colleague for suggestions on how they avoid particular errors, for instance removing the word "pubic" from your dictionary will ensure if you type it instead of "public", SpellCheck will pick it up.
An apology will usually solve the problem as most people understand that mistakes are made and recognize that an apology takes a lot of courage and professionalism and they usually respect that, but if you are an HR assistant and sent the job offer to the wrong candidate, then more damage control will need to be taken, your boss will need to be told and the consequences could be more serious.
Recently I sent about 80 invitations to a dinner by email and then had to send 25 more to another group of people requesting a meeting. After putting the message for the invite in 80 times, by the time I got to the next set of emails, well, I put the same message in the email regarding the dinner so it did not match the letter I attached. My damage control was to re-send the message with REVISED in the subject line and then follow up by calling the 25 offices and speaking to the EA to explain the situation. It was easily rectified as they all had access to their bosses email acount and they all most definitely understood. You can be sure if they make a similar mistake and call me, I will do the same for them and have in the past. In this case, after I had done my damage control I mentioned my error to my boss, but along with the remedy so he saw that I recognized what I had done and took the appropriate action to fix it.
I remember once an assistant from a law firm sent a fax to my boss by mistake and it had the legal advice they were giving their client, which my boss wasn't supposed to see, so she immediately called and asked me to trash it and that she would send the correct fax. I waited to get the correct fax and when I saw it was a simple error of putting the wrong fax number in, I had no problem ripping up the other fax she had sent. Many times I dealt with this same person and we were always helpful to each other and she would bend over backwards if my boss needed to meet with hers to accommodate the request. How we handled it turned it into a positive working relationship.
If you have handled it to the best of your ability, righted the wrong and told the people you needed to tell, then you need to move on. I have seen people who kept kicking themselves over a mistake, but failed to learn from it and all that did was affect their self esteem and confidence in doing the job. If you dwell on every mistake too much, keep telling your boss how dumb you were for doing it (and therefore reminding them about the mistake), that will not help the situation and will likely make your boss have less confidence that you won't make the mistake again. I believe if we tell ourselves something for too long, we inevitably start to believe it about ourselves. The quicker we deal with it and move on, the better for everyone.