抱歉 (bào qiàn) is a more formal and sincere way to say 'sorry' compared to 不好意思 (bù hǎo yì si), which is used for minor social blunders or embarrassment. Use 抱歉 to express genuine regret for causing trouble. Note that 遗憾 (yí hàn) means 'regrettable' and describes disappointment about a situation, not a personal apology.