My short definition:
Managing systems for other people to use.
Most of you have probably done at least some of the things system administrators routinely do, especially install and configure operating systems and applications and do other system management tasks on your own computers. What is different about system administration is the focus on the needs of other users of a system, and not just your own. This focus is what drives many of the issues that become important in system administration: stability, reliability, and security; change management and documentation; service to a user community and ethical considerations.
Mark Burgess's textbook definition:
. . . system administration is a branch of engineering that concerns the operational management of human-computer systems.
Burgess is an academic researcher with an interest in developing more formal and scientific approaches to system administration. He has written possibly the first paper proposing a scientific theory of system administration: On the theory of system administration. In short, his theory is the idea that system administration is fundamentally the process of keeping a system at or near some optimal state, based on various objective metrics for what can be considered optimal.
Another recent, and quite different, characterization of system administration is stated in Alva Couch's article From tasks to assurances: redefining system administration. He says
System administrators do not "perform tasks" or "apply expertise" but, rather, "provide assurances."
Rather than trying to define system administration as some large set of possible tasks, he attempts to characterize system administration as providing assurances about the availability of computing services.
So-called "cloud computing" has also significantly changed the nature of system administration recently by focusing on building larger, more distributed systems; using automation tools to manage system properties that were once managed by hand; and often outsourcing hardware management to external providers.
A sysadmin's description of his job (local copy)
A much more cynical sysadmin's idea of what makes a sysadmin?