A true apology requires three things
1. Admit that you were wrong and that you're sorry. Really own up to what you did — or failed to do. For example, "I jumped down your throat and berated you mercilessly when you didn't get that report done on time. I was wrong to treat you that way and I am sorry." Sadly, most labor attorneys will advise you not to say you're wrong to anyone, because that might lead them to have something they can use to sue you. If that is the case, you may just need to stop at saying you're sorry. (And while in matters of the business and legal world, perhaps you shouldn't admit you were wrong, in matters of the heart with the people you love, always say it. It's that one thing they need to begin to forgive you.)
2. Show them you understand the effect it had on them. For instance, "And when I did that, and wouldn't let it go, I think I made you feel cornered and probably anxious — and maybe even panicky." You don't need to jump to conclusions or make assumptions about what they must be feeling or thinking; just try to really put yourself in their shoes.
3. Tell them what you are going to do differently in the future so that it doesn't happen again.For example, "Going forward, when I'm upset about something you have done or failed to do, I'm going to pause and ask myself, 'What is the outcome I want from speaking to you? In all likelihood it will be for you to just fix what needs to be fixed so you can get the results that both of us want. I will calmly speak to and maybe even with you instead of at or over you." This reassures them that you will truly try to change your behavior in the future — not just keep apologizing after every blow-up — and hopefully ends the conversation on a more positive note.