(1) I'm sorry, (2) It was my fault, and (3) How do I make it right?
Very often, people miss out (3). (3) brings out the sincerity in the apology.
Some people even miss out (2). The apology then becomes superficial, or what we call in Chinese, 敷衍.
A good promise, in my opinion, also has three parts.
(1) I promise ... ... (and mean it), (2) Actually remembering and completing the promise, and (3) If, for some unforeseen circumstances, you cannot complete the promise, apologise (for good apologies, see above) and offer an alternative.
An if you are unable to do the above, then please refrain from making any promises. The greater the expectations, the bigger the disappointment. It's cruelty to bring others to the high grounds of hope and let them fall flat on their face.