For many British people, apologising is a default reaction to
life's little irritants. If someone barges into you, treads on your
toes or spills your drink, it is considered quite normal for the
victim to mutter "sorry". This is clearly illogical, but for many
British people it is an ingrained response. The urge to apologise
for other people's actions is clearly misplaced; constant, needless
apologising devalues the currency, and will lessen the impact of a
genuine, heartfelt mea culpa.
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.
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 apologies 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.