There are as many definitions of leadership as there are leaders and followers. (And, there will be even more definitions of leadership if one considers it an outcome, not an input.) Consider this sample of definitions, all colored by the perspectives of their authors:
Leadership is the capacity to translate vision into reality.—Warren Bennis
Leadership defines what the future should look like, aligns people with that vision, and inspires them to make it happen, despite the obstacles.—John Kotter
A leader is one who knows the way, goes the way, and shows the way.—John Maxwell
I start with the premise that the function of leadership is to produce more leaders, not more followers.—Ralph Nader
Leadership is the art of getting someone else to do something you want done because he wants to do it.—Dwight Eisenhower
A leader is a dealer in hope.—Napoleon Bonaparte
Leadership is lifting a person's vision to high sights, the raising of a person's performance to a higher standard, the building of a personality beyond its normal limitations.—Peter Drucker
A leader has to be somebody who's getting people to do things which don't seem to make sense to them or are not in their best interest—like convincing people that they should work 14 hours a day so that someone else can make more money.—Scott Adams
Leadership is about service to others and a commitment to developing more servants as leaders. It involves co-creation of a commitment to a mission.—Robert Greenleaf
You don't lead by pointing and telling people some place to go. You lead by going to that place and making a case.—Ken Kesey
The point is that what correlation one might find between teacher leadership and student scores will necessarily be a function of what definition one gives to leadership. Peter Drucker's definition could help but any choice would have to be justified, if only because some might deem it self-serving to tie the definition of the independent variable (or cause) to the predetermined dependent variable (or effect).
The simplest way to do this is to run a Spearman correlation between the two variables. As Hector noted, the Spearman correlation coefficient can be defined as the Pearson correlation computed on the rank transformed variables.
I'm confused because my data is not paired. I mean that not each x variable has its own y variable. Each teacher skills( ordinal variable) is related to many student scores (continuous variable).How can I study the correlation?