The verb to apologize is s'excuser. Excuser (or pardonner) means to forgive.

There are many ways to apologize, depending on the seriousness and on the person you got in front of you.

Minor trouble and known person

For example, you hit someone you know:

• Pardon !
• Excuse !
• Excuse-moi ! forgive me!

Minor trouble and unknown person

For example, you ask for the time...

• Excusez-moi... Auriez-vous l'heure s'il vous plait ?

Serious trouble

• Je suis vraiment désolé(e)... veuillez m'excuser.
  I am really sorry... please forgive me.