There are many ways to say “I am sorry” in French:

The quickest way would be “pardon” (“sorry”), “pardonnez-moi” (“forgive me”), or “je suis désolé” (“I am sorry.”)

But you can also say “je vous demande pardon” (“I apologize”), “je vous présente mes excuses“, “Je vous prie de m’excuser” (“I beg your pardon”), or in an even more formal way, “je vous prie de bien vouloir m’excuser“, or “veuillez accepter mes excuses.“

“L’erreur est humaine; le pardon, divin” (“To err is human, to forgive divine”), says the French maxim today, borrowing it from the famous English poet Alexander Pope, who himself had quite a few reasons to be “pardonné” (forgiven), considering the irreverently “satirical” aspect of his œuvre littéraire (literary work.)