An international trademark does not exist. However with the Madrid Protocol, companies are able to file one application for trademark registration with a group of countries. The Madrid Protocol, which includes 90 countries, is primarily a treaty that enables businesses to file a trademark application for a group of countries.
You can also file an application with the Community Trade Mark (CTM) system which provides protection for trademark owners in all 28 countries in the European Union.
Alternatively you can file a separate application for trademark registration in each country that you intend to promote your products or services in.
Regardless of what approach you take to file outside of the U.S., each individual country has their own requirements regarding trademark use and registration.