1.) The very first two words to apologize. Learn them. Use them. Start with them before you start with anything else.
they are: (everyone together now) I'M SORRY. You dont necessarily have to bold it out like that but it's amazing how those two words can start an apology ball rolling. Even if you dont THINK you need to/should be sorry. Still say it and start with it.
2.) Continue with a reticent point on you were not intending to harm feelings, or to make offense, and stepped on toes and the like. You don't have to butter it up or schmooze it with insincerity but do be simple and to the point.
3.) Regardless of who is in the wrong or who is in the right, don't spotlight the original offense as fault or blame. Maintain a mistake was made, repeat that you're sorry that mistake and miscommunication caused the grief. Do not try to be "right" (or "righteous/self righteous") EVEN if you MAY be.
4.) If you can swallow a little pride, stroke the ego of the friend you may have crossed and are trying to apologize and make amends. Nothing says "meet me halfway" by trying to give something to smile about or feel good about again. Again, use sincerity and be honest and real.
5.) Know when to be sorry and 'shut up'. The thing about apologies is, you can make them awesome, but don't over-make them. Because even I'll admit there's a line where you have to 'stop being sorry about it'. Let whoever you wronged process it and either accept it, or not.
6.) Accepted apology!? Great! Don't make the mistake of going right back to the original fault and/or trying to constructively criticize. There is a time for that.. but not directly after an apology. (Ie: we know sometimes the truth hurts, but the time for the truth hurts is not during an apology attempt)
7.) Accept that on occasion an apology might not be 'enough'. But don't let that prevent you from trying to make one just the same to re open doors of friendship. Its very true that a good many don't know how to accept an apology either. But that's another entry for another time.
Utilize the above, and a few tactics of your own, and amaze a lot of people, and feel a little better about yourself, in having learned the tact, and tactics of making good apologies.