What are the site inspectors' duties?

His main responsibility is to make sure that the work is carried out to the client's standards, specification and schedule which is prepared by the architect or an engineer employed by the client.

As its name indicates he is responsible to check the execution of works as per standards and specifications.
Also some times he is responsible to verify the quantities at site.

to check and ensure that the job executed complies to the specs and standards

To check whether contractors & sub contractors are working as per schedule planning,quality of work,safety on site,to check the bills of quantity submitted by them as per the completed work & so on

To ensure that the work is progressing as per the designs & specifications under better safety environment.