To protect your trademark internationally, you must file for trademark registration country by country. Protection is dependent on territory and only granted in the country where the trademark has been effectively registered.
There are a few regional agreements, such as the European Union Intellectual Property Office, that allow you to protect your trademark in multiple countries through a single application. However, worldwide trademark registration cannot be obtained in one go, and registration must be obtained by applying before the trademark offices of all desired countries.
The Madrid protocol for trademark registration process, which is managed by the World Intellectual Property Organization, offers a system that can facilitate the process of registration of an international trademark.