Math PhD application process
There are two main models of organizing doctoral programs:


The American system: graduate students proceed through a combined master's and doctoral program, usually starting with coursework, then taking some sort of "qualifying exam" to become doctoral candidates within the department. Admissions are handled at the department level.
The "German" system: doctoral candidates are employees rather than students. Candidates for a position must apply directly to the group of the professor who is sponsoring the project. There is no coursework phase, as applicants are expected to have already completed their master's degree. (However, they can "audit" courses if needed.)


I would suspect that these represent the "poles" of the spectrum; most other systems I'm aware of fall somewhere in between. [Note that even the German system is adopting some features of the American system, and is thus starting to become a hybrid.]