Admitting you made a mistake is difficult for anyone. However, refusing to face up to being wrong or causing a problem can take a toll on your reputation, relationships, and work life. As difficult as it can be, admitting the mistake can allow you and others to move on.
Talk to the person affected. If you can't tell them, at least talk to a close friend.
Start with an admission you were wrong. For example, say: "I'm sorry I started that rumor. It was a really hurtful and childish thing to do."
Apologize for the consequences. For example, say: "I'm sorry that I've ignored your opinion. I see now that I really hurt your feelings."
* Only apologize if you are sincerely sorry. An insincere apology will hurt the other person more.
Explain yourself if you need to. For example, say: "If I had known that Jane Doe had the problem under control, I wouldn't have gotten involved and created that mess."
Explain why it won't happen again. For example, say: "That was really foolish of me. Next time I see a situation like that, I'll ask for details instead of jumping to conclusions."
Give the other person space to respond. Depending on the situation, the other person may take time.
Let it go.
If it's a minor mistake, or not very serious, make a joke about the incident. It shows that you're comfortable with yourself and can get over minor difficulties. For example, say: "Next time we have an important client, I'll have to have cheat cards like a newscaster. Then I'll remember get name right!"
Don't pressure someone for forgiveness.
More serious conversations should be in person. Less serious issues can be handled via phone or email.

You can't control others responses. However, you can make sure YOU move on.
Learn from your mistakes. Assess what went wrong and how you can prevent it in the future. If you forget the incident, you're likely to repeat the mistake.