There are many reasons for wanting to get a website shut down. Perhaps you find the content offensive or degrading, or maybe the site is involved in questionable practices and you want to see it banned. However, just because you don't like a site doesn't mean that you can just make a call and have it shut down. In fact, unless the site is local and is doing something illegal, it can be very difficult to have it shut down.
Determine if the site is breaking the law. If the site is doing something that's blatantly illegal such as displaying child pornography or selling body parts, then contact your local authorities and ask how you should proceed. If the site is not hosted in your country then there may not be anything they can do.
Examine the site to see if it is violating intellectual property or copyright laws. If so, contact the site owner and see if he/she will agree to remove the offending material. If not, then you may wish to contact a legal professional for information about sending a DMCA (Digital Millennium Copyright Act) notice. However if you really want the site shut down, and not just have the material removed, then you may wish to contact the web host.
Contact the company who hosts the website. You can find the company by performing a WHOIS search on a site such as WHOIS.net. Enter the domain name of the site you want shut down. This search will tell you where the site is hosted. Once you have this information, go to the web host's site and examine its terms of service. If the site you want shut down is violating any terms, notify the host and ask it to shut the site down.
After performing a WHOIS search, you will see a listing of nameservers. Whatever domain is listed is where the site is hosted. For example, if the nameservers listed are ns1.website.com and ns2.website.com then you would visit website.com and examine their terms of service to see if the site you want shut down is in violation.
If you contact your local authorities they may tell you that you have to contact the authorities in the state or country where the site is hosted. If that's the case then you can find that information by doing a WHOIS search at WHOIS.net Sometimes you may have to perform multiple WHOIS searches. For example, if you perform one search and the domain listed in the nameservers is not a working site, you should perform a WHOIS on that non-working domain and then visit that domain's nameserver address.