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:
Minor trouble and unknown person
For example, you ask for the time...
- Je suis vraiment désolé(e)... veuillez m'excuser.
I am really sorry... please forgive me.