If you are not the one actually running the router or a DNS server on your network, it would not be easy to see what websites others are going to, and without being system admin more than likely not legal depending on your country without authorization from your company.
Most routers have an option to log to a certain computer. If your system admin can forward the logs to your computer then you can see which ip's your computers are accessing and filter the logs. Other than that, you can host a DNS server that logs lookups that all computers on your network attach to.
Ettercap is capable of doing what you want, but without official authorization from your organization, this would probably be considered an intrusion and could get you arrested.
Most of the internet keeps information to those who need to know it...your sysadmin with the router, your internet provider along with their domain name services, and the website admin. Most websites have stats that the owner can look up to see who is on it, but will not share this information with the general public.