Service encompasses pretty much everything that an academic does outside of teaching and research. This includes advising, serving on university and departmental committees, reviewing papers, writing letters of recommendation, organizing seminars and conferences, serving on professional boards, etc. The impossibility of doing it all allows academics to decide what types of service they are going specialize based on their interests and abilities.
I have spent the last three years heavily engaged in university level service. I currently serve as the president of United Academics of the University of Oregon, and I am a past president of the University Senate. Before that I was the Vice President of the Senate and the chair of the Committee on Committees. All of these roles are difficult and require a large investment of thought and energy.
The reward for this hard work is a good understanding of how the university works, who to go to when issues need resolution, and who can be safely ignored.
As important as the service I am doing, is the service I am not doing. I do routinely serve on academic committees for graduate/honors students, etc. I decline most requests to referee papers/grants applications, and serve on no editorial boards. The national organizations for which I am an officer are not mathematical organizations, but rather organizations dedicated to shared governance.