Not a lot has been written on the topic. Most people seem to use verbal apologies. I am uncertain if this is truly as popular as it seems, because they seem to fumble and be unsatisfied with results fairly often.
In any case, it is essential to apologize in a way that makes your partner feel apologized to, because people can have different preferences. I myself prefer to find out what went wrong and how to prevent it from going wrong again (if possible), and in some cases a concrete component is very helpful for restoring the relationship to its usual state. I'm much less interested in verbal apologies as they rarely change my feelings or events, except for a few things like "I'm sorry, I was wrong, I should have listened to you and will try to do so in the future." (After "I'm sorry" and its derivatives, "I was wrong" is possibly the most useful phrase for relationship repair.) YMMV.
You will sometimes see concrete apologies mentioned as "making restitution" among the languages of apology. Regrettably most of these talk about it without describing the steps in detail or explaining how to work through it.
Most of the detailed versions are religious and not particularly helpful. Those from business angles are sometimes better. Here is a general list of steps for an effective apology that includes restitution.
"Making amends" is another way of framing a concrete apology. Most of these are aimed at recovering addicts (Step 9).
A special case is what supervillains call Step 9 1/2 -- making amends indirectly when when making amends directly is impossible or inadvisable. Making amends is more concrete than merely making apologies. It is better done directly, but when this is unfeasible, it can be done indirectly. In Terramagne-America, the legal system relies a great deal on penumbra restitution, because victims often want nothing to do with perpetrators. So the perpetrators pay their debt to society by doing something of use to the same category of people they harmed. Ideally, the penalty should connect to the crime in some way, rather than being arbitrary. For example, someone who caused the death of others could register as an organ donor.
There are a few generic examples of making amends, though.
Some useful phrases:
* How can I fix this?
* How can I make it right?
* How can I make it up to you?
* How can we repair our relationship?
* What would help?
* Would you like it if I ...?
Some useful categories:
* If your mistake cost the other person time or happiness, which is typical, then do something which will give them one or both of those things.
* Do one of their chores that you know they dislike.
* Do a chore that you normally avoid because you dislike it.
* Do something they often ask you to do, that you rarely if ever do because it's unpleasant and/or you aren't very good at it.
* Hire someone else to do a thing which will benefit your partner.
* Buy something that will save them time, otherwise make their life easier, or prevent future conflicts.
* Give up an activity you had wanted to do in favor of a conflicting one your partner prefers.
My go-to example is washing the dishes. I hate it and I'm not great at it, so my partner typically does it. If I feel that I have made a mistake, I can do a sink of dishes and he will immediately recognize that as a concrete apology, whether or not I have said anything.
Some people, mostly Ask culture, prefer to be asked what they want. If you guess, they may not appreciate it as much. If you offer several options ending with an open slot, however, they may respond favorably. Other people, mostly Guess/Hint culture, prefer that you know by observation and extrapolation what they would want and do it without needing to ask. If you have to ask they will feel like you don't know them or care about them. Also asking for things is hard for them, which is the opposite of helpful in this instance. It is generally advisable to know what your partner and close friends like and dislike, so you can offer tempting options or just go do something suitable, whichever they prefer.
So then, my steps for concrete apologies go something like this:
1) Consider the problem. Do you feel that you have made a mistake, or is someone else telling you that you have?
* If you feel that you have made a mistake, think about what personal standards you have violated and how you can get back to where you belong in your own eyes.
* If someone else is telling you that you have made a mistake, listen to what they say.
** If you agree that you made a mistake, listen further to ascertain what they would like you to do in order to fix it, and how to prevent a repetition.
** If you disagree, this is going to get a lot messier. You have to decide whether to give up your own interests in favor of the other person and do as they wish next time; or whether to admit that you two have a conflict of interests and let them know you will make the same choice again. If so, they will probably be quite displeased, as most people become angry when they cannot get what they want. But sometimes goals or needs are simply opposed. Sometimes you can still compensate by giving them something else they want, in the interests of preserving a relationship, or fairness, or just not being a dick.
2) Apologize. Most people appreciate a verbal component. Tastes differ regarding its content and importance.
3) Discuss options and make a plan for the future. In most cases you will wish to avoid a repetition of this incident if you can. Some things are straightforward to fix. Other things may take much practice before you can do better, though. Careful analysis and planning will help.
4) Make a concrete apology. Concrete means something you make, buy, or do; not just words. Make supper when it is not your turn. Buy a timer so you don't lose track of something you're supposed to do. Shovel the snow off the sidewalk while someone else relaxes. If you are not good at thinking up things on the spur of the moment, make a list of things when you hear others talking about what they wish they didn't have to do or would rather you did for them.
If you are at the a loss regarding what you could do to fix a given error, then consider the state of your relationship bank account. This metaphor addresses the amount of positive energy each person contributes vs. the amount of trouble they cause. I think it's an exaggeration to say you have RBAs with everyone you meet, but certainly you do for all the people you see regularly enough to recognize each other and recall the nature of your interactions. Most humans seem to have a monkeysphere of around 100-250 people. That's about how many RBAs a typical brain can handle. Here are some tips on maintaining a healthy RBA. Whenever you feel clueless, you can just make random deposits of whatever your partner typically likes. Most people will eventually respond favorably to this tactic, although the length of time required depends on the scope of your mistake. As a general rule, you can also make people more willing to put up with your inevitable mistakes by deliberately doing nice, helpful, thoughtful things on a regular basis -- the same way you save money so you can replace a blown tire without overdrawing your bank account.
Here are some of my poems that deal in mistakes and apologies. They include a wide range of offenses, different apology languages, and levels of fluency in apologizing. Think about what they do, what works and doesn't work, and what you might do in similar situations.
"Ten Ways to Use a Spoon"
"A Moment of Atonement"
"The Bravest and Most Lucid Thing"
"The Most Powerful Force Available"
"The Willingness to Surrender Ideas"
"With a Tender Hand"
"What I Can Ethically Read Aloud"
"You Sure Do Change the Future"
"To Try Making Things Right"
"A Voice from Beyond"
"With Your Heart Open"
"The Feast of Saint Valentine"