As european (Dutch) and webbuilder:
This applies to EVERY website targetting europeans, no matter where you host or where the company originates.
• This website has plenty of info about EU legislation on cookies
Might be nice to know, you now longer need to place the noticifation if you only use Google Analytics (you used because GA uses a cookie to check for returning visitors) and cookies specific for the website. Because of this, most small common websites don't need a niticifation to the user.
The reason GA ís allowed, is because they don't track you from site to site, only if you come back. This is considered acceptable because it is basic information which is useful for a webmaster and not privacy invasive for visitors. These cookies are available for the visited domain only and therefor seen as first party.
FYI, it's called cookie law, but this doesn't only apply to cookies. Session.storage and similar functionalities fall under the same rules. Everything that tracks users for the purpose of tracking users.