Measurement of a user's contributions to the site, and the extent to which the community "trusts" that user with extra privileges.
Reputation is the greasy oil and lubricant that powers one side of the Q&A Stack Exchange engine. (The others being the content and the users.)
Reputation is a rough measurement of how much the community trusts you; it is earned by convincing your peers that you know what you’re talking about. Basic use of the site, including asking questions, answering, and suggesting edits, does not require any reputation at all. But the more reputation you earn, the more privileges you gain on Stack Exchange.
For more information on how reputation impacts what functionality is available on the site, as well as how you can lay your hands on some, read the gory details in How does "Reputation" work? and be further enlightened.