I thought I'd give you some advice on how to handle these things. You may not always come out as the "winner," but you'll come out looking sane/smart/less-of-a-bitch if you follow these rules. So here we go.
1. Do not call the other person names or use any demeaning adjectives towards them. This includes (but is not limited to) calling them a bitch, and asshole, stupid, an idiot, a dumbass, a douchebag, ect, ect, ect.
2. Do not bring past issues into the argument. You need to discuss what is going on right at that moment, not what's happened before.
3. Keep calm, or at least the appearance of being calm. Yelling, screaming, cursing, caps, and other things that express anger just get you no where in solving the issue. Not to mention these things normally make you look ridiculously stupid.
4. Don't bring your friends or theirs into the issue. Sure, you can ask your friends for advice as I'm sure they are, but if you have to send them to talk for you then you look like a wuss. Fight your own battles.
5. Your first instinct does not need to be to cut them off or ignore them. The first solution attempt should be talking the problem out. If you're reading this, you should be able to handle mature conversation.
6. If the other party doesn't respond to you, try again in a day. If they're mad, maybe they need time to cool down? After that point though, you might as well just give up or wait until they can grow some balls/put on their big girl panties and talk about it.
7. If you find yourself putting a lot of effort into working things out, but they're just shutting you out completely and not really working it out, give up. This shows a major flaw in their character and you should just move on.
8. Don't make the dilemma public. Third-parties tend to be biased without fact and do not think before they speak and this can lead to a lot of hurt. Especially if you haven't given the other person a chance to speak.
9. And speaking of fact, before you start causing shit, make sure you know both sides of the story. It's wrong to make someone look/feel bad when they've done nothing wrong. Don't just jump to conclusions.
10. Try not to burn bridges. I understand that in some situations you really have no choice but to do so, but if this is a little thing don't do it. I'm not saying you should talk to them, but don't just tell them to go fall off a bridge and never look your way again. Creating enemies is never a good idea; I don't care who you are.
11. Don't talk down to them. Even if you are smarter, you should try to put yourself on their level. Maybe they'll understand the issue better. Please, don't make your goal be to make them feel stupid.
13. Always be honest, and remember: Forgiveness is the best policy.
There are 3 main types of apologies, but only 2 of them are acceptable. I'm going to address all 3 of them here.
1. Taking the blame.
9 times out of 10, this apology will be accepted because the other party will usually be happy to have any blame relinquished from them.
This is an apology in which you say that you did x and you now feel apologetic and/or guilty for doing so. The other person takes no blame, and you hope to get a clean slate afterwards. It's pretty straight forward.
(i.e. "I'm really sorry for everything I did. I was being a really shitty person and I hope you can forgive me.")
2. No one is/both are to blame. It was just a big mistake.
These are probably the hardest to do without messing up the wording somewhere. However, with the right audience and enough thought, these apologies can be some of the most heartfelt. This is the apology to use when you truly want to make up with a person.
These apologies typically have few words and never make an accusation towards either party.
You don't take blame for the situation, and you also do not place it on the other party. You are basically apologizing for the negative emotions and the whole situation even taking place.
(i.e. "I'm sorry about what went down, and any xyz emotions you may have felt because of this. I hope we can move past it.")
3. The other party is to blame.
Honestly, if this is how you feel, why are you even apologizing??
This is an apology where you put the blame on the other party, usually through passive aggression.
i.e. "Hey, I'm sorry you felt that way. I know you were probably just confused and it caused a big misunderstanding. It was all a big mistake, and I hope you'll be able to move past it."
Do you see how accusatory that is? Using sentences like this in an apology usually won't end in a positive result because no one likes feeling blamed. Even if they should feel guilty, you can't make them. If they've done something wrong, they will have to come to that conclusion themselves.
Also, don't expect them to return an apology. Only apologize if you truly feel that you should. Not because you want to fish an apology out of them.
Don't feel bad if your apology is rejected. You took a step to fix things, and they didn't want to take that step. That's their character flaw, not yours.
It is NEVER too late to apologize. Never forget that. A year later, or even 20, it's still not too late.