Michael Feiner wrote a book called The Feiner Points of Leadership (get it?). The book has 50 laws, but they mostly say the same thing - Leadership is about managing relationships. A better subtitle for the book may be "50 examples of leadership", but it doesn't matter - it's a good book, and if you are a leader (or want to be one) I think reading it is worth your time.
As a manager, I have obvious responsibilities to manage my relationships with my employees - as well as try to manage their relationships with each other. I also need to manage the relationship with my own boss. Every manager has these same relationships, but not every manager is necessarily a leader. I don't think you can be a leader by only managing direct relationships (i.e. your direct reports and manager) - that's just management. The challenge in leadership is managing all of the other relationships you are responsible for. Leaders must manage relationships with peers, peer teams, customers and many others to be successful. Good leaders build trust and credibility in all of their relationships, and use those attributes to make everyone perform better. Influence with out Authority is something that leaders need to be able to do. Making people to do stuff just because they work for you is never going to be a key to leadership success.
Of course, I'm not saying I do all of this stuff, but I do think about it. I know that I don't work enough with my peers, and I know that I need to do a better job managing relationships with our customers, but it's something I think about often, and something I hope I can continue to get better at.