For me, the term "self-centered" applies more so than selfish. Self-centered without being narcissistic, which is relatively harmless. I realized this not too long ago, but I used to condemn myself for my nature. To think that I didn't care about anyone because of how self-centered I am. Why does society have to place a taboo on something that is quite obviously natural? It can be bad like anything else if there is an excess, and it does not need to be condemned. If we were all more honest with ourselves and the traits that make up human nature, I believe this would be a better world. It is not a bad thing to care about oneself, and learning to love myself is something that I'm doing right now through extensive therapy. But I digress.
I have spent most of my childhood engrossed in my own mental landscape, with little of anyone else to care for. My self was and is my center. Hence my usage of that word. It caused me to develop a heavy self-preservation impulse, and caused my entire ego to expand into the size of the worlds that I created. That's not to say that I'm egotistical in the traditional sense. It is just that I find it very difficult to understand others and their emotions, whilst who I am and what I feel is very clear to me. I am ridiculously self-aware.
Further, everything I do must directly relate to me, or else, I become bored. And the complaints of others often seem very irrelevant and stupid to me as well. (Although the reason for that is because I have a mother who enjoys crying wolf a lot, so I have little tolerance for other people talking about their distress.) When I roleplay, I must be a character who is similar to me or has traits that I wish to emulate or I just don't feel very much of anything. In media, my favorite characters are those who I can see myself in. My entire existence is centered around relating things to myself, finding the similarities and differences between me and everything else. I constantly spend loads of time drawing representations of myself as well. Pokesonas, fursonas, or just my plain old human self. And fine-tweaking them eternally to feel exactly right in expressing me.
Yet because of the way I am, I have to overcompensate for my lack of understanding. I force myself into thinking about how I would feel in every situation that another is suffering, relating it directly to me and... well, if I become too close to a person, it becomes difficult to tell the difference between myself and them. It's the only way I can care about something easily, because of my generally detached nature to everything. So you could say that in a self-centered way, I become selfless. I attempt to apply the suffering of another to myself, and I end up acting with an excess of compassion. It becomes unhealthy at this point. That's something I'm working on.
A lot of the times as well, I'll disregard my own feelings over another person's, because they are outside of me and what I feel isn't as important due to the fact that I have more control. I can control my emotions, but I can't of these strange chaotic beings who exist outside of my world. And thus, I end up becoming overly concerned about their feelings, because I don't want to be yelled at or punished for being a bad person.
Basically, I am self-centered yet selfless simultaneously.