Most everyone has done something wrong or has had something done wrong to them, which has basically necessitated the words of "I'm sorry" from the party that has done the wrong. In consequence of that wrong, for some people, even making an apology is difficult and can even be heart-wrenching as they wrestle with their conscience or perhaps their humility; on the other hand, for the vast majority of people, saying that one is sorry is not only fairly easy, but appears often to be basically a pleasantry that really doesn't mean much of anything, except perhaps as a general acknowledgement of some sort of generic failure or wrong.
In point of fact, an apology, in many cases, whether acknowledged or accepted or not, really doesn't mean a whole heck of a lot, when the words are as simple as "I'm sorry," and nothing else is said. For the very first question of the party that has been apologized to, might well be, what exactly are they sorry for? After all, the person declaring that they are sorry, might internally really be saying that they are only sorry that they were caught or found out, or it could be that they are basically being forced or compelled to say that they are sorry because of some perverse power structure, just to keep the peace; so that, in actuality, in many ways, their apology can be quite hollow.
Then there are those that do apologize and also provide an explanation behind the apology, in which typical explanations follow one of these two paths; of which the more common seems to be the pleading that though they are apologetic, there are obvious extenuating circumstances, that created the situation, and only because of those circumstances, justified or not, did they commit the act that necessitated the apology. The other path, is an abject apology, specific addressing the act that created the need for the apology, and owning up to their responsibility to it, without any "ands", "ifs" or "buts", and placing the blame on no one else, even when some of the blame may, in fact, lay elsewhere.
While a very good and well meaning apology most definitely has its place, the bottom line, is that those that are truly apologetic, without exception, need to go beyond making an apology, but also need to take the next logical step, which is to ask for forgiveness from the party that they let down and hurt. So that when anyone apologizes but also requests forgiveness, they have gone beyond a mere soliloquy into desiring an actual dialog. After all, when it comes to apologies, there are at a minimum, at least two parties involved, and the party that has been harmed has the right to not only have their say, but also retains the power as to whether such will or will not be forgiven at that time and place.
That is to say, not every apology is accepted on the spot, nor should every apology be accepted on the spot. Further, the request for forgiveness is not necessarily something that can be provided at a moment's notice, but usually requires some sort of reflection and contemplation, for not every act, lends itself to forgiveness, for some acts are so heinous and traitorous, that forgiveness is not going to occur at that time and moment. Yet, the very act of asking for forgiveness, starts the process of healing in motion, for those that will not admit to their error, are the hardest to forgive, for they apparently do not know or will not acknowledge that they have done wrong; and wrongs not acknowledged by those that are the perpetrators of them, are the hardest and deepest wounds to heal, by those so offended.