A sincere apology should always be offered when your actions have had a negative impact on other people. Even if you do not fully understand why someone is so upset, respect their feelings, and accept that your actions are the root of the problem. Don’t pass the buck, or use your apology as a way of blaming someone else. Take full responsibility for your actions. On the other hand, resist the (very English) urge to apologise for other people’s actions. If someone barges into you, a muttered “Sorry” is misplaced. Constant, needless apologising devalues the currency, and will lessen the impact of a genuine, heartfelt mea culpa.
An apology will be much more persuasive if you acknowledge the fault: “I’m sorry I was so late” is more specific than a simple “I’m sorry”, and actually recognises the other person’s grievance. Never temper your apology with accusations or insinuations: it will negate its impact. If you have committed a real faux pas consider sending a handwritten note – but only after you have offered a verbal apology, otherwise it will look like cowardice. If you are offered a genuine apology, acknowledge it graciously and accept it. The urge to elicit grovelling self-abasement is both childish and offensive.