Apologies apparently have six common elements:
– an expression of regret
– an explanation of what went wrong
– an acknowledgement of responsibility
– a declaration of repentance
– an offer of repair
– a request for forgiveness
Generally speaking, the more of those you include, the better it works, but the the research below finds the most effective elements are taking responsibility and offering reparations. The least critical element was a request for forgiveness.
It's worth noting this study only looked at written apologies. Given other research about how the impact of what you say depends far more on body language and tone than actual verbal content, consider how to look and sound contrite. (Note: sincerity helps.)
Here’s the Best Way to Apologize, According to Science
Sorry not sorry? Here’s how to fix that.