I've held onto grudges for way too long because the other person couldn't apologize properly. And I've definitely fumbled apologies myself.
A real apology takes responsibility. No "I'm sorry you feel that way" nonsense. Acknowledge what you did, explain why it was wrong, and commit to doing better.
Forgiving Too
Also learn to accept apologies. Holding onto anger only hurts you. If someone genuinely apologizes, try to move forward. We all make mistakes.