Reputation is leaving "points" for other members based on the posts or threads they make. These points can be positive or negative, and you can choose to leave a comment or not.
vBulletin has this to say:
Users gain and lose reputation based on how their posts are scored by other forum participants. Users with the ability to affect reputation, will either give or take aways points by approving or disapproving with a post's content.
How much reputation you leave depends on the date you joined, your number of posts, and your reputation level. More of these make you give or take more reputation points.
You give reputation by clicking the

button next to someone's post.