When you see someone who seems to exist for the sole purpose to care for and be kind to others, or someone who gives up things so another person can have a necessity they lack, the word that comes to mind is humility. A humble individual does things without regard to personal gain.
To be selfless is one aspect of humility Being humble means you do something nice for someone, like open the door for a woman struggling with three children or giving up the last seat on a crowded train, without expecting anything in return. You are satisfied because of the act itself and not from the breathless "thank you" from the mom or the loud "you are so kind!" from the person to whom you gave your seat.
A person who expects something in return for any act of kindness, no matter how big or small, and becomes upset when they receive nothing, is not showing humility. A truly humble person is fulfilled without the need for a thank you.
Have you ever said, "I'm sorry" just because you felt like it was required and not because you actually meant it? That is not a way to show humility. It is nice when you apologize or admit you made a mistake. However, it only truly means something to the person you are speaking to when it is sincere and honest. When you admit the error of your ways and make a concerted effort to show that you were wrong, that is a humble act. It shows more strength in your character to be humble and apologize _ and mean it _ than to try and defend your actions.
Being humble means you do not need lots of fanfare when you successfully tackle a major feat. Think of the professional athletes who jump on top of a scorers table after they dunk a basketball or do a goofy dance after a touch down. Those are not humble acts. A humble response after scoring the winning point in a game would be to walk over to the opposing team member, shake hands and say, "Great game." The crazy actions may get more immediate attention, but the humble act will be the thing the teammate and opponent appreciate for years to come.