There are two types of leadership: Institutional, and Behavioral.
Institutional leadership is what we imagine normally when we think of the concepts of "leaders" or "leadership". These institutional leaders (officially) affect the actions of others directly and assume responsibility/accountability for the decisions of those under their influence. This may happen to varying degrees bases on competencies. One needn't be in a "leadership position" to be this kind of leader, but there must be some organization within a group (even if unspoken).
Behavioral leadership acts in such a way so as to encourage others to emulate him/her. This is also referred to as "leading by example". No organization is needed to be a behavioral leader.
The people most frequently recognized as leaders are those who exhibit the characteristics of both categories. Of course, actually good leaders from either category are hard to come by. But why?
First, people don't typically aspire to be "leaders". Many aspire to be in control, own a business, or be revered, but that's just ambition-- which is only very loosely related to leadership. Few understand the burdens involved and thus never learn to shoulder them. Be it living the virtuous life (or "walking the walk") or being able to admit without concern "yep, that error came from my team", people assume that either is too difficult.
Similarly, there are everyday leaders that exemplify the institutional and behavioral leader but refuse to take on the title because they assume it requires position and rank.
Today, we're kind of mixed up in the realm of leadership because the ambitious have retooled the term to their own cause while the actual leaders frequently attempt to evade visibility. Still, we would be better off if we could more consistently shine the light on real, everyday leaders and use them as role models for leaders of every type in the future.