Emotional courage is the ability to do something even when you are afraid.
There are several examples of emotional courage in the book. For instance, Atticus is defending Tom Robinson even though it is not popular.
Scout, simply by the nature of the work, every lawyer gets at least one case in his lifetime that affects him personally. This one's mine, I guess. (ch 9)
Atticus also uses Mrs. Dubose as an example of emotional courage, because she stands up to her own fear as well as her physical addiction when she weans herself off of morphine.
I wanted you to see what real courage is, instead of getting the idea that courage is a man with a gun in his hand. It's when you know you're licked before you begin but you begin anyway and you see it through no matter what. (ch 11)
The key in this case is that you keep fighting, even if you think you are going to lose or you are afraid you are going to lose.
Emotional courage is being brave by ACTING brave even when you don't FEEL brave. Atticus exemplifies this because he defends Tom Robinson with everything he's got, even though he is almost certain he will lose.