Apologizing to your boyfriend's mother is many times the hardest thing you can do. It means swallowing your pride and making things right with her. Consider writing a letter to apologize to her. Choosing to write a letter to apologize gives you the opportunity to fully express yourself without interruption. You could also add a small token of what she means to you with the letter.
Share from your heart when writing the letter to your boyfriend's mother. Most people respond positively when they sense you are being genuine. While it will be important to share what she means to you or how sorry you are, be sure to avoid exaggerating those points. Exaggeration or flowery speech can come off as desperate or manipulative. Keep it simple. Authentic language grabs the heart of the reader.
Express respect towards her as the mother of your boyfriend. Honoring who she is, regardless of any differences, will likely warm her towards you. Recognize all that she sacrificed for him and everything she poured into him. Give her credit for her influence in making him the man that he is today -- the man you love. Share the qualities of your boyfriend you love the most and thank her for instilling those qualities in him.
Take responsibility for your actions and choices without any excuses or minimizing what you did. Avoid shifting blame to her in any way. Even if you feel she was partly to blame for what happened, that is not your place to point out. By taking responsibility you move the relationship forward and demonstrate you value the relationship. It shows maturity on your part and will perhaps open the door for her to take responsibility for her part. This may be painful, but you will gain respect and trust in a very strategic relationship.
Acknowledge how your actions affected her. This takes the apology a step farther and raises the level of genuineness. Seeing it from her perspective requires a depth of humility few people are willing to experience. This process is healthy for everyone. It also, typically, wins the heart of the person you are communicating with. She may share with her friends how thoughtful and humble you are, and open her arms, heart and home to you forever.