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This is the most basic way of saying "I'm sorry" to someone in French, and you should use it in most formal situations.  The phrase "Je suis" means "I am," and "désolé" means "sorry" in the form of an adjective. Write "désolé" if you're male and "désolée" if you're female. Regardless of gender, pronounce the entire phrase: zhuh swee day-zoh-lay  You should use this phrase when apologizing for something serious, apologizing to strangers, apologizing to elders, or apologizing in a professional or other formal situation. If you need to apologize for something very serious, you should say, "Je suis vraiment désolé" (male) or "Je suis vraiment désolée" (female).  The term "vraiment" means "truly" or "really." When inserted into the usual apology, you're essentially saying, "I am truly sorry." Pronounce the entire phrase: zhuh swee vray-mohn day-zoh-lay When you're in an informal situation, you can drop the "Je suis" portion of the phrase and simply state, "Désolé."  Essentially, you're dropping the words "I am" and simply saying "sorry" instead of "I am sorry." As with the formal version of the phrase, you should write "désolé" if you're male and "désolé" if you're female. Regardless of gender, pronounce the apology:day-zoh-lay  Use this phrase when issuing a quick apology to friends or other close peers. Ideally, you should only use it when the offense was relatively minor, and should opt for the formal "e suis désolé" after major offenses.
Say "Je suis désolé" in formal situations. Add "vraiment" for serious matters. Offer "désolé" in casual situations.