here are commonly two ways to apply for a Ph.D. programme.

Route 1: You have to check the webpage of the universities that you intend to study, browse to narrow down your search to the department and course of study that you want. Check the research areas of the Professors that interest you and then make contact with them via email (most common method of communication).  This basically means that you make independent research and write your dissertation (which has to be published) under professorial supervision. Most times, available PhD positions are advertised on the websites of the Professors or the corresponding departments. Please note that the Professors or department can offer you monthly stipends to support you during your studies.

Route 2: The second way to make application for a PhD is through a graduate school, Research training Groups or International PhD programmes. These are structured PhD programmes. That is, they are organized in such a way that there is a time frame (usually six semesters on the average) within which you have to complete your PhD studies. They are also funded, which means that you get monthly stipends to support yourself.

More information can be obtained from the Professors, the Graduate Schools and Research Training Groups that  you are applying to.