To me, the biggest offenses are often those so painful to the one who committed them that it may be almost unbearable for that person to even allow himself to talk about it (including apologizing). Whenever I've been "the offendee" I haven't really wanted the person who did something and didn't mean to do, and maybe didn't know any better; apologize to me or do anything else to make up for it.
For me, a simple acknowledge that the "offense" happened is all I ever need. For example, once someone simply said, "I'm sorry. That's not what I wanted...". (that type of thing). Someone else once said, "I'm sorry. I didn't know."
It became an over-used and super-corny joke soon after it "was out there", but in the movie "Love Story", do you remember, "Love means never having to say you're sorry?" I think "getting it" remains important, but sometimes, I think, when people have a very strong bond, sometimes they do know how sorry someone is and truly understand that the person didn't intend harm.
I know what it's like to do something that hurts someone even when that was the last thing I'd ever, ever, want to do (and most of the time, people don't mean to cause the hurt/harm they do to those close enough to be "more hurtable" than, say, strangers).
My "thing" has always been that all I'd like to get from someone who has done something seriously damaging/hurtful in my life is for me to know that the person "gets it" - that's all. I'm fine with understanding that it's so painful to the one who committed the offense he/she can't talk about it. (Maybe he/she can't even allow himself to think about.) I just need to know - without a doubt and once and for all - that s/he "gets it".
As for strangers/outsider and others that I'm not close to (maybe a professional/business relationship), then I switch to the "talk is cheap" thing and want compensation - and maybe firing, suspended or terminated professional licenses, and written apologies.
Also, talk may be cheap but it's still the best way to "connect" genuinely. It shouldn't be underestimated.
With any more minor offenses; to me, a sincere and simple "I'm sorry," (sometimes with a simple explanation, perhaps) is generally all I want/need. :)