Everyone has their own way of defining leadership. Here's mine.
I think that leadership, as a concept, should be compared to management. Management is performed whenever someone deals with complexity and leadership is performed whenever someone deals with change. They can be performed at the same time, and usually are in some proportion, but this distinction makes them easier to think about. They are not concepts which can be separated; they are two sides of the same coin.
If we think of actions like they were all mixed up together and baked into a brownie, then we will have a very strange mental image. However, we will also have a useful image. The center, the majority, of the brownie is the same soft consistency. This majority would represent how most of the things we do are only in response to complexity. The edges of the brownie, which are harder and chewier, are the minority of actions which are in response to change. It's not such a metaphorical leap to imagine the inside of a brownie as all being the same and the edge of a brownie being the part that has to "deal" with the change from brownie to non-brownie, which is why it's a little bit different from the rest of the brownie raw materiel.
The proportion of soft brownie to chewy brownie varies from situation to situation. Some jobs are characterized as "Leadership" jobs because one tends to either seek out or be confronted with more change than usual. This idea could be represented like so:
The reason this metaphor is useful is that it illustrates a principle of the relationship between management and leadership. They really cannot be found in isolation from each other. You can't bake a brownie that has no edges and you can't bake an edge that has no brownie. Additionally, the center and the edge of a brownie are made of the same stuff (actions in case the metaphor is still too vague). We tend to label a job as either management or leadership because of the relative proportion of edge to area. So, a marble would be an example of a "management" job because it maximizes management decisions and minimizes leadership decisions.
This could be compared to a radiator which maximizes surface area vs volume, just like a "leadership" job would maximize leadership decisions and minimize management decisions.
So, when people try to describe leadership as fundamentally different from anything else they are forgetting that leadership is just the edge of the brownie that ran interference between the brownie and the outside world. It is still the same stuff, it has only acquired a different consistency due to its being exposed to change. Management and leadership always exist together, but the proportion sometimes changes.
Management, then, is when someone is dealing with something that is well understood. We like to keep things the same for as long as possible because we need to be able to rely on something. That means that after a while the thing that has always been the same is so well understood that every problem has a documented solution. A person only needs access to this accumulated knowledge and they can maintain the status quo indefinitely. Leadership, however, is when someone is on the leading edge dealing with change. To maintain an area in which things do not change, someone has to be out on the edge figuring out how to deal with the inevitable new problems.
There is no documentation which allows someone to deal with a new situation because nothing can be documented until it has first been dealt with. So, leaders have to basically guess the future, which is why leadership is so hard to understand. People keep trying to approach leadership the same way they approach management, as a complexity problem instead of a change problem. Another way to think about it is that there is a fundamental difference between dealing with something you can see and something you can't.
The latter picture, of the shadow, is what leaders are expected to focus their attention on. They are supposed to learn how to interpret things with no context and no support. Is it any wonder people are so confused about what they're supposed to do? On the one hand leaders are supposed to inspire confidence. On the other hand their job is basically voodoo, and their success is largely dependent on luck, which does not inspire confidence.
This brings me to a key point, which is what leadership is not. Leadership is not management. A manager who delicately applies some creativity to deal with some workplace drama exercised leadership. A leader who carefully practiced his speech and incited the masses to follow him exercised management. The methods for inspiring people to do things is relatively well documented and is becoming more so every day. Therefore, inspiring people should be characterized by the preponderance of management inherent in the action. Leadership, on the other hand, is dealing with a situation without any supporting information. It is inherently impossible to define HOW to 'do' leadership, then, because every situation is different.