What most people already know about being a script supervisor is that they, well, supervise the script. "But what exactly does that entail?" one may ask. The paramount job of a script supervisor is to maintain a daily log of the day's shots and their relation to the script during the production process. They act as the chief continuity person on set and act as the on-set liaison to the postproduction staff.
Essentially, the script supervisor is the editors' best friend. They're practically the only ones on set who have the postproduction staff's best interests at heart. They keep track of every shot and make sure it'll be able to be cut together later in the production process. We love our script supervisors!
Getting a little more into the details of the job, a script supervisor's foremost task is to maintain of log of all shots taken on set. This log has every bit of information an editor could possibly need. It has shot numbers that match up with the script, slated shot numbers, comments from the director or DP (usually explaining important details of what to use/not use), continuity information, the tape/reel number, camera settings, the date and time, production statistics, etc., etc., etc. Any and all questions postproduction staff may have should be answered by this log. If there are any questions left unanswered, the script supervisor didn't do his/her job and will most definitely be fired...ok, maybe not fired, but someone will definitely be slightly upset.
Along with the continuity notes on the shot log, the script supervisor must also have keen eyes on set so as to call out any errors while shooting. This can include anything from an actor's hand movements from shot to shot to a prop placement error. Clearly, they can't interrupt a shot just to point out an error, but instead will relay their notes onto one of the directors so that they can correct them in future takes. To assure for consistent prop/set placement and design from day to day (you never know what fool will decide to move something), the script supervisor cross-references with the continuity stills photographer daily. They need to make sure they'll have easy access to any continuity stills during production so they can quickly double-check anything they need to without costing the production any time.
A script supervisor who's particularly good at their job will most likely have some level of editing experience, or at least a general understanding of how it works. It makes for a much better judge of what can and what cannot be cut together (continuity). Speaking for anyone who's ever worked in postproduction, we love our script supervisors. We really, really do.