What is a "very good higher education degree" when applying for a German PhD?
Basically, you need to have a master's degree. If that master's degree comes from:


A German university
A university in any member country of the Bologna Process 
A few other countries considered "equivalent" but not participating in the Bologna Process, such as the US, Canada, Australia, and Japan


then it is almost always automatically accepted. On the other hand, if your degree is from another source (such as India, China, Iran, Africa, etc.), or is from a German Fachhochschule, then the degree must be certified by the university as being at the same level as a German master's degree before you can be admitted. Also, after your admission, you may be required to take some additional courses to establish your candidacy (although this is usually on the order of two to three courses during your first two years).