Although it's hard to say that there is a "typical" process, most PhD admissions are handled both an administrative staff member (often with a title like Graduate Program Administrator or Officer) and by an admissions committee made up of faculty.
The administrative staff will generally filter applications. For example, if applications are incomplete (e.g., missing grades, test scores, or recommendations), the administrator will often remove these applications from the pool. If the department has firm requirements (e.g., minimum TOEFL scores or GREs), the staff will often remove applicants who do not qualify from the pool.
This smaller pool will be reviewed by the faculty. In many departments, this will be by the graduate admissions committee who may then reach out to individual faculty members who seem like potential advisors to strong applicants. In some other departments, the files of students may be sent directly to potential supervisors by the administrative staff.
Faculty will usually be able to see the unfiltered pool but will not often look at the applicants who have been filtered out by the staff based on the most objective criteria of application completeness or eligibility.