The first thing you need to do is ACCEPT RESPONSIBILITY. Admitting that you made a mistake and letting others know you're to blame makes people more receptive of your apology. APOLOGIZE to the people you offended. Be genuine and forthright. Without going into great detail, explain the reason you made the mistake and reassure the person(s) it won't happen again.
Try your darndest to MAKE THINGS RIGHT if you can. Take any opportunity to reverse the effects of your mistake. Unless you own a DeLorean with a “flux capacitor”, you cannot erase the mistake you made, but striving to improve the situation is crucial to making amends.
You may actually have to BACK OFF a bit because some people feel wounded when they’re the victim of someone’s mistake. Trying to push for acceptance of your apology may leave them feeling resentful. Give them time to recover and re-establish trust with you.
After proving through your actions that you have LEARNED YOUR LESSON, try to regain a sense of normalcy in your relationships (hopefully "normalcy" is good). The other person's demeanor and body language will indicate when you've made satisfactorily amends. MOVE FORWARD and put the incident behind you.
Remember to let the other person(s) set the tone for your interactions until things get better. In trying to put it all behind you quickly, you may become overbearing. That’s the last thing you’ll want. TIME heals all.