Truly accepting an apology
means that you have thought it through, acknowledge that your friend made a mistake, and embrace the fact that your pal is sorry. While you might still be a bit angry, you are on your way to letting go of that in favor of a solid friendship once again. Saying that you accept your friend's apology means that you are willing to put an end to the argument in such a way that it will not continue to be an issue going forward.
When your friend tells you they are sorry, embrace their apology as a way to move on from an argument. Do not keep bringing up the issue or continue reminding your friend about what happened.
More about apologies and forgiveness:
"We all make mistakes. I accept your apology."
"After thinking it through, I realize this was just a misunderstanding and that you didn't mean to hurt me. I accept your apology."