We have all received a quintessential non-apology: I'm sorry you were insulted. I'm sorry the truth hurts. I'm sorry you misunderstood the joke. Although these all include the phrase "I'm sorry," they may come off as insincere and ultimately place the blame on the person receiving the apology. With a heartfelt apology, you admit you were wrong and take responsibility for your actions in the hopes of alleviating the other person's hurt and making amends. Sometimes, though, you may want to smooth things over despite the feeling you did nothing wrong.
Consider whether or not what you plan to apologize for is actually your fault. If you have done something wrong, the first person you have to admit it to is yourself. Although your pride might be less wounded by avoiding responsibility, you will make the situation worse by offering a backhanded or disingenuous apology. Even if a certain situation affected your actions, you are not necessarily blameless. The point of an apology is not to negate or detail such circumstances, but to allow healing to occur by honestly and clearly admitting what you did do wrong.
Determine if an apology really is called for. There are times when you will have the urge to apologize even though you didn't do anything wrong. Women in particular have a tendency to apologize for all kinds of things they have no control over. Dr. Charlie Glickman gives the example of arriving late when you left the house late versus arriving late after hitting an unexpected traffic jam. It's reasonable to apologize in the first case, but the second one isn't your fault. In this situation, he suggests changing the wording. You can say, "I regret I was late." Here you express condolence but don't take the blame for what happened.
Determine if an apology is called for to smooth over a social or work situation. In an article on Jezebel.com, ethics author Lauren Bloom uses the example of apologizing to an in-law, even though you may be right, in order to relieve family tensions. Bloom also notes that there are plenty of times when another person has a very different experience of a situation than you did. You might be wrong and not even know it.
Be careful with your phrasing so that you don't step into the terrain of a complete non-apology. In the case of the in-law, Bloom suggests coming from a place of sincerity -- truly wanting to make amends to alleviate the suffering of your spouse. You can say something along the lines of, "I'm sorry. I see that what I said caused hurt in the family. I won't do that again." If you aren't willing to go that far, try to account for the other's experience by saying something like, "I had no intention of causing you any pain, and I'm still not sure what I did was wrong..." and continue from there, speaking from a genuine place of empathy.