To apologise is "to express regret for something one has done wrong".
It is different from expressing sympathy for somebody's distress, although the words used may be very similar:
- "I'm sorry I upset you" (an apology);
- "I'm sorry you were upset" (expressing sympathy without admitting to any wrong-doing).
It is particularly important to bear the distinction in mind when "apologising" on behalf of somebody else. If, for example, a patient complains about one of your juniors, and you "apologise" without first establishing whether your junior had acted appropriately (but the patient was upset anyway) or whether they had e.g. acted insensitively, you are prejudging the situation (by accepting that they had done wrong without first checking whether they had), and your junior would have grounds for a complaint against you.