A Script Reader is quite likely the first person to read your script once you have submitted it to a competition or a film company. You can think of them as gatekeepers, as they will go through your script and score individual components. The higher the score, the more likely your script will progress.
Script Readers also have another role. They can read through your script and produce a report for you. These reports can range from an Overview Report (one or two pages) to a Development Report (10 pages of in-depth pros, cons, and ideas for your script).