You can choose any of the pillars to be your focus, but you need to brainstorm first and then write an essay about how the other three pillars play a part in the primary one. For example, if you say leadership is the most important, then you need to prove that and then explain how your scholarship has affected your leadership; how your integrity has shaped your leadership; and how your service record reflects your leadership skills. Well-rounded individuals display all of these qualities which is what makes them excellent candidates for NHS. The point to learn by writing the essay is that having only one or two of those qualities is nice, but that all of the those qualities are enhanced by the others, making the candidate a better person.
Because this is an essay for National Honor Society, you probably want to reveal a little bit about yourself...
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within it, rather than writing a dry and objective essay. (I assume this essay is part of your application.)
The "dominant" pilar to me would suggest the one I most easily excelled in. Then I would talk about why that pilar was my strength, and how the others were either byproducts of such a strength, or characteristics necessary to have this strength.
So I would probably choose leadership. I happen to be a pretty natural leader and always have been. I would write one paragraph about how my character affected my leadership ability, one paragraph about how my grades were affected by my leadership, and vice versa, and finally, a paragraph about how service is a natural by-product of being a good leader.
Do you see where you have a lot of room to make this essay personal, and that there isn't really one correct way to write it?
Anyone who is a good leader or scholar would exhibit integrity and service. Scholarship is built on honor (academic integrity) and service (we believe that our work is important to the world.) The same could be said for the other pillars, as previous posters have pointed out. So you really just need to make up your mind which one you think is the most important, and tell how all the others are essential to it. It is also always a good idea in essays like this to show how you have modeled these qualities in your life.
A strong character is certainly the most personal (and I think the most important) aspect of an individual's makeup. The other aspects you mention--leadership, scholarship and service--will all follow a person who has high moral integrity. Leadership skills, for example, are built upon others' view of a person, and a student without a solid moral center is not likely to be viewed as a true leader.
You need to choose which one of these is most important to you. Integrity means you value telling the truth and being a good person. Character means you value friendship and being a good person, and you follow your own moral code. Here is a brochure that will be useful.
http://www.nhs.us/portals/3/content/55490.pdf