I'm right with my colleagues so far. Be direct, tell them what you're sorry for, and ask their forgiveness. My reminder is for what happens after the apology is made. When it's something that really matters, it's almost always easier to say you're sorry than for someone to truly forgive you. And once they do, it's easy to assume you will be back on the same footing as you were before "the incident." If you truly mean it, you must also be prepared for the consequences of your mistake. Forgiveness may not mean a renewed relationship or trust. Just a reminder that even a direct and sincere apology may not be enough to restore what was lost--but it's still the right thing to do.
If you are truly sorry then it is best to be up front and honest. I have always found that honesty is the best policy. When people are up front and apologize then they are also respected more for doing so.
One thing that I would try to avoid is apologizing via facebook, texting, etc. I notice that so much communication takes places through technology. People are losing their face to face communication skills and this is something that we definitely do not want happening.
Above all I would say it depends on what you have done as well as other matters! The best way to think about doing it though is bravely admitting that you were wrong and saying sorry directly to the person. You can then explain yourself more if you feel the need. Often, if the other person really feels that they have been wronged by you, this is one way to patch up the relationship or to build bridges between you. An open admission of guilt and a direct apology will mean a lot to them. It is important however not to try and excuse yourself or to pretend that it wasn't really your fault - a true apology just consists of you admitting your own responsibility and sharing your sadness and feelings of guilt about having done whatever you have done. Good luck!
I find that the direct approach is usually the best. The apology must be sincere, or the person you are apologizing to, who is probably defensive since they have been offended in some way, will be able to tell and your apology will fall on deaf ears.
Face to face is always the best way to offer an apology. Explain how you screwed up, admit that you were wrong, and let them know it will not happen again. Use the words I'm sorry, or I apologize. If you are too nervous to apologize face to face, or you don't think they will meet or listen to you, write them a note saying the same things. Avoid apologizing by text, or in an IM or a post, as you do not want them to miss your meaning or think you don't mean it.