In the movie The Breakfast Club, what role does leadership play? What types of leadership are used? In what ways does leadership help the group achieve their goals?

The two big leader figures in The Breakfast Club are Mr. Vernon, the assistant principal assigned to check in on the students in detention, and John Bender, the delinquent student. Mr. Vernon represents society at large, while Bender is a rebel who questions the social order, often angrily so.
Mr. Vernon's idea of leadership is essentially small-scale tyranny. He is not interested in the students as individuals and instead perceives them as the social personas they wear as representations of their cliques. For example, he thinks Bender is destined to be a criminal loser because he does not see why Bender is the way he is, nor does he even guess that his lashing out might be from alienation and pain.
Mr. Vernon tells the students not to move or speak for hours and write an essay. All of these rules are pretty arbitrary and are unlikely to actually make the kids think about what they did to get into detention. He is ultimately contemptuous of the people he's supposed to be leading and is therefore a failure as a leader, despite his good job and respectability.
In contrast, Bender questions the rules, his disdain for authority stemming from his suffering at the hands of an abusive father. He inspires the other kids to start probing why they have boxed themselves into particular social roles. Bender is the student who cares the least about conformity or respectability, and his open disdain for the rules coaxes the others into opening up to one another about their shared fears and dreams.
In this way, The Breakfast Club's notion of ideal leadership is represented by Bender. Bender allows the other students to be themselves. He does not just expect everyone to conform to a particular "type" and never imposes rules on the others. Unlike the proper authority figure, Bender inspires real, positive change in the students.

In John Hughes's coming-of-age masterpiece The Breakfast Club, each student—each from a different social background—proves, in a sense, to be a sort of leader. Forced to deal with one another and the innumerable amount of differences and quirks they all possess, the students hold significant influence over each other.
Bender, the most obvious leader of the pack, is a misfit who helps the more uptight students unwind. He rebels against Vernon by messing with the doors, resisting Vernon's commands, making a ruckus, and other forms of tomfoolery. While the other students initially loathe Bender and find his behavior moronic, they soon join in, even defending him when he comes crashing through the ceiling during a botched escape attempt. Bender opens the students up by offering them marijuana, which opens the students up emotionally to one another. While the other students tread lightly out of fear of further discipline, Bender's supreme indifference teaches them that conformity is not always applaudable.
Ironically, Andy, the football hero, has little to no influence on the others. He scraps with Bender, flirts with Claire, and generally ignores the outcasts. As the film progresses, Andy's true character is revealed: he is only a leader externally. He puts on a show to keep his popularity and to earn the respect of his father. Andy confesses to the others that his athletic ambitions are non-existent—he only plays the role of a football hero. When Andy attempts to keep the students calm, obedient, and quiet, he fails; Bender takes control and serves as a proverbial wrench in the machine.

It is ironic that the sole authority figure in The Breakfast Club is someone who does not show much in the way of leadership. Mr. Vernon does not want to be there that Saturday morning any more than the kids do. He makes $31,000 a year, has a home, and is at the stage of his teaching career where he really does not care anymore. He hates the kids; they are a constant source of vexation and he does not want to have anything to do with them. To Vernon, kids are just those annoying creatures who get in the way of the smooth running of Shermer High.
So Vernon does not lead. He does not inspire the kids with anything but loathing and contempt. He simply lays down the law and leaves the kids to their own devices. Bender, however, unexpectedly shows leadership skills, albeit of a more unorthodox variety. As well as being a perennial screwup and troublemaker, he also has great charisma, which he uses not just to challenge Vernon's formal authority, but also to inspire the other kids to rebel.
And their rebellion is not just against Vernon, but against their parents and the society they represent. The unlikely figure of Bender provides the catalyst for the kids to explore and express their deepest, innermost feelings. They are not about to start acting like Bender when they return to school on Monday morning; but he has led them to get in touch with their true selves. This is something that only Bender's charismatic leadership could ever have achieved, and certainly not the formal, rules-based leadership of Mr. Vernon.