A leader is a person who steps up to lead a group towards a common goal.
Leadership is a skill set that helps you naturally be a leader. Leadership skills include: leading group conversations, being a spokesman when needed, being able to take the blame and work through the issue/problem, engaging your group in some or multiple ways, being able and knowing how to run meetings (aka planning), being able to delegate without becoming a dictator (or just a dick!), etc.
As a natural leader, I find that its important to be able to take the blame, know how to not single people out in a group setting, be a spokesman, say "no", and take on most of the work load.