Why do we apologize?
Apologies are the emotional needle and thread we use to continuously mend our relationships. They're also powerful vehicles for growth.
Good apologies are an absolute necessity in any long-term intimate relationship. Without the ability to both admit mistakes and forgive others for their own missteps, we don't stand a chance in building the type of durable, lasting relationships we long for.
What is an apology?
We can define an apology as an attempt to rid oneself of blame, either proving no wrong occurred or, as is more often the case, acknowledging fault and making amends.
When a disagreement or tension arises, we always have a choice.
We can assume a fault genuinely existed and pave the way for repair, or deceive ourselves (and our partners) to obtain some kind of easy "resolution," opening the door to repetition.
A genuine apology can only be made when we acknowledge our fault in an occurrence. It appears obvious, but it's easier to apologize due to social/partner pressure or fear of loss than to actually acknowledge our fault. This isn't apology, it's appeasement. It does little, if nothing, to encourage growth or repair a rupture.
While expedient, this false apology erodes the truth in a relationship and in ourselves, as we repeat the same mistakes. Over time, we lean on appeasement more to hastily pave over discord, even as our partner's trust in us begins to creak and crack.
True apology incorporates our role in the conflict. It requires we take responsibility not for our own behavior and our reactions to the situation. Action and emotion together.
In "The Art of Living," the stoic philosopher Epictetus says:
If it is our feelings about things that torment us rather than the things themselves, it follows that blaming others is silly. Therefore, when we suffer setbacks, disturbances, or grief, let us never place the blame on others, but on our own attitudes.
Epictetus
In other words, if what torments us is anger, insecurity, frustration, fear, etc., it makes no sense to blame external forces. Instead, we need to look within and examine our own unpreparedness to deal with these emotions.
Only when we grasp this unpreparedness can we feel remorse.
Unpreparedness is a literal gap in consciousness — which originates from the Latin conscire, meaning "to know something with oneself," or "to possess oneself."
We can only recognize our fault when an occurence violates what we know about ourselves, and our values. In other words, if we know ourselves poorly, it's harder to recognize our faults and easier to blame others. This may be difficult to read, but if you find yourself constantly wondering why you are being treated a certain way, or not understanding why you are being criticized, it's likely you do not know yourself adequately.
Knowing others is intelligence; knowing yourself is true wisdom. Mastering others is strength; mastering yourself is true power.
Lao Tzu
What makes for a good apology?
A good apology is the acknowledgment of a mistake without the expectation of forgiveness. It's an act of both humility and accountability, as we identify our limitations and commit to overcoming to them.
A good apology is expressive. It's an outpouring of true remorse and shame.
A good apology is empathetic. It takes into real account the other's perspective, and their feelings about our actions.
A good apology is genuine. It aligns our actions, or future actions, with our values.
In a good apology, we don't expect a favorable reaction from the other person. Instead, we show ourselves and the other our remorse, the shame we feel weighing on our conscience. We don't merely explain our own pain, how awful we feel, and beg that we be released. We own our shame, and use it as a real driver in our effort to make things right. After we have done that, we accept the consequences, which are not ours to control, and move forward.
While most of us harbor shame for far too long, shame is a crucial ingredient to apology because it shows, perhaps more than anything else, that we are vulnerable. There are real limits to how far our love, patience, respect, etc., extend. This makes us fallible. It also makes us human.
As therapist John Bradshaw put it, shame is the "emotion that lets us know we are finite." We misapply shame when we burden ourselves with it. Shame is a tool for growth. Shame leaves us when a certain level of growth has been achieved.

How to apologize?
Before an apology, one must ask a series of questions: "What did this person feel when I acted this way?"; "What could I have done differently?"; "Can I actually be different or just hope to?"
Often, we are accused and criticized for actions that another person feels we could have done differently, but this demand doesn't always align with what we are capable of offering.
No one would expect a different action from a blind person who bumped into a vase and broke it; their ability to see is limited. Similarly, we have our mental and emotional limitations, which our ego's often misjudge. We shouldn't hide in a feeling of hopelessness for ourselves, just as we shouldn't believe we are boundless in our ability to change at the drop of a hat.
The question of "What could I have done differently?" requires balance and emotional maturity. We must resist the impulse to justify. Most of us naturally want to deflect blame onto external circumstances or the other person's behavior, especially when emotions run high. "I wouldn't have done X if not for…", "You deserve it after you did Y." We'd rather evade responsibility than risk overwhelming ourselves with the work it would take to make a real correction.
Let an honest personal assessment of what you are actually capable of guide you on your path. A patient man can genuinely apologize for not having been more patient. A man with one leg can't apologize for not going on a run. You can't give something you don't have. In this case, you should save your apology—which won't be genuine—and spend that time expanding your own capacity in the area you need to grow.
When to apologize?
There is an optimal window for an apology. It ought to take place after acknowledging the mistake and before the other becomes resentful.
If we apologize without acknowledge, or a sense of responsibility, it isn't genuine. Either we disregard the other person's feelings, or what we think and feel doesn't aligned with our words.
If we apologize after the right moment, it becomes irrelevant. Negativity takes root in the other person. The apology loses its power of restoration.
When someone suffers because of our actions, and we care about that person, we need to stop and reflect on ourselves as soon as possible. A bond has been broken and needs to be repaired.
Think of a relationship like a patchwork quilt, stitched together with time and commitment. When we make mistakes, we risk undoing those stitches. When we opt out of authentic apology, or we appease instead, these loose threads grow, and more of the stitching comes apart. There is a chilling effect when a heart turns to stone. Frustration becomes anger. Anger becomes resentment. Resentment becomes bitterness. Bitterness becomes contempt. Finally, the damage exceeds any chance of repair and the quilt falls to pieces on the ground.

What if forgiveness doesn't come?
Even if we are genuine, empathetic, and expressive at the right moment, there is always the possibility that our actions will not be forgiven. People are people. Situations are different. Some mistakes are graver than others. We must be prepared to accept the consequences.
He who does his best, does all that can be expected of him.
Helena P. Blavatsky
By withholding forgiveness, the other person deprives us of an opportunity for redemption. Make no mistake, they deprive themselves too, choosing the path of anger over reconciliation.
This deprivation weakens, if not destroys, our redemptive efforts within the relationship, at least for a time. It does not, and cannot, destroy the power of redemption itself.
If we cannot redeem ourselves in another's eyes, we are left to redeem ourselves in God's eyes, which you can define in anyway you choose. This is the real work. The act of forgiving yourself is the hardest task of all, especially when no one else will.
We forgive ourselves by recognizing our flaws and seeking to work on them. Through this, we acquire power over ourselves and our past. We know we are different. We do not need any one else to tell us so.
In the end, only forgiving ourselves can free us from the burdens of our conscience. To fulfill our potential, we can't afford to walk heavy, bound in the chains of our past. There is an open path ahead of us and a bright light beyond that. There's no magic wand to heal every wound, but we can choose to walk this path, choose to do the work, and choose to keep our heads held high as we go.
"No one can go back and make a new beginning, but anyone can start now and make a new ending."
Chico Xavier