As a first step you should find a professor who accepts to supervise your doctoral project at FAU. We recommend having a closer look at the relevant websites of the field you are interested in. The site http://www.fau.eu/university/organisation/faculties/ gives you a first overview for starting your investigation. Additionally, you might also find within our structured doctoral programmes suitable projects and supervisors.

You can contact the potential supervisor directly, sending your CV and research plan. Please be sure that you tie in with former publication or current research projects from the potential supervisor: generic application and applications without a point of contact won't be accepted. 
The best opportunity to get in contact to a supervisor is certainly a personal conversation, for example at a conference, if you cannot meet her or him here in Erlangen / Nuremberg directly.
He or she must agree in supervising your planned research work before you can start doing your doctorate at our University.

Information on the further steps is gathered on our website on how to start your doctorate at FAU and in our Doctorate Guide http://www.graduateschool.fau.eu/guide.