empathy

Again, that comes down to how it's communicated. For example, of someone's parent dies, dropping by to say, "I'm really sorry you have to go through this. I've been there and know how much it sucks. If you'd like to talk, I'm always available."

That takes the form of an apology, while not apologizing for anything and instead recognizing, validating, and offering help with their feelings.

That's obviously and extreme example, but I think it demonstrates my point.

Another more realistic example in a relationship would be someone getting frustrated about their partner's busy work schedule. "I'm sorry that I'm not around as much as you'd like, and please understand that I also miss you when I'm gone. I also really enjoy my work, and this is the balance that gives me the most time with you without shirking my responsibilities at work."