What is the role of a graduate coordinator or graduate chair (I have seen faculty called both) of a department in admissions ? Do they make the final decision or have the final say in all decisions ? How much power do they have in admissions ? This is in North America. The graduate coordinator or graduate chair in question is a faculty member of the department.
(In the context that the person in question is a faculty member, not the support staff for that faculty person...) I have the impression that this varies considerably from department to department, and also varies depending on the person, on the current dept head, and other politics within the dept.
Let's call the person the "DGS" ("director of grad studies", in math).
At my R1 univ in the U.S., in math, the DGS is appointed by the dept head, and needs approval for policy and funding choices, if nothing else.
The DGS has variable influence in choice of the grad admissions committee, which is ultimately done by the dept head. Nevertheless, the DGS is literally the person who would make admission and funding offers to grad student applicants.
Yes, the general task of sifting through all grad applicants is distributed among the grad admissions committee, organized by the DGS. The DGS collates all those reports, and makes decisions.
In some cases, individual faculty (on that committee or not) have particular applicants that they advocate for, which is typically conclusive in the applicant's favor. But this is usually a relatively small fraction of all admitted+funded applicants.
(As is typical in the U.S., the vast majority of math grad students are funded not directly via individual faculty grants, but as Teaching Assistants, or to some degree by Fellowships of various sorts...)
So, yes, the DGS has considerable power in grad admissions (and funding), but is answerable to the dept head, and needs to not annoy toooo many faculty.
(I think in most places the DGS does not literally, directly control the funds that pay teaching assistants or fellowships...)
I assume in this answer that the graduate coordinator is an office person, rather than a faculty member. If they are a faculty member, then this answer doesn't apply, though in that case some of the responsibilities I describe may still fall to that faculty person; besides that, they are likely to have just as much influence as any other faculty member of the admissions committee
Admissions decisions are made by an admissions committee consisting of several faculty members.
"Graduate coordinator" in my experience refers to an office support staff person. They have expertise in navigating the academic bureaucracy and can be a huge asset in sorting out funding issues and improving the experience of students in the program, but they do not have decision-making authority on admissions.
They may assist the admissions committee by sorting through and excluding some applicants based on particular hard criteria; for example, GPA or TOEFL requirements, having a bachelor's degree or progress towards one. I suppose they could exert some power if they "accidentally" let an application fall in the trash or not get distributed to an admissions committee member, but these would be unofficial ways to express that power and not fitting to university policies.
That said, a graduate coordinator can exert quite a bit of soft power. If an applicant is rude to them, it may very well get back to the committee. I have heard stories where an interviewing student was incredibly rude to a graduate coordinator and this pretty much torpedoed their admissions chances, not because the coordinator had any power but that the committee heard about their behavior and didn't want that person on campus.
If there are problems with your application, it's probably the graduate coordinator who will help you out. For example, there was a recent question here from someone who had submitted their graduate application very early, and they now had updates to their CV: the graduate coordinator would probably be the person who could get their updated CV included in the materials the committee reviews. Same thing for late-arriving reference letters: if there is any discretion to include something arriving late it's likely the graduate coordinator will have some influence.
In my experience (United States), the director of graduate studies is the head of an admissions committee. Did you mean this role when you said graduate admissions coordinator? Committee membership can vary year to year based on faculty's service commitments. This committee often has some rubric to evaluate applications, and each member is responsible for evaluating X number of applications. When it comes to decision making, the committee convenes and discusses the applications to narrow them down to a list of who they would like to admit/extend offers, and sometimes a waitlist is formed. I would say the graduate director has some influence, but each faculty member (based on tenure) has influence in who they might admit because they are looking at students who they might work/do research with. Is this at all helpful?
The graduate coordinator is usually in charge of the paper flow. The coordinator may be a faculty member if the program is small, or a deputy to the graduate chair.
If the coordinator is a faculty member, she or he will usually be on the admissions committee, and will have one vote (although in principle and in cases of emergencies I believe that our grad coordinator has the power to recommend admission without consultation, but that's never been used). If the coordinator is an assistant, this person will support the committee and may or may not be present at committee meetings. The coordinator could be in charge of dealing with emails, distributing and keeping track of information on the availability of funds or scholarships, for instance. She or he would know the rules and regulations of the program, and would basically liaise between the department and the Faculty of Graduate Studies to make sure that applicants meet the eligibility criteria, that the department does not overoffer, etc.
The role is not necessarily limited to admissions: they could be in charge of nominating chairs for thesis defences or project presentations (or whatever other formal requirements of the programs). Often they will sign the departmental paperwork to be sent to the Faculty of Graduate studies indicating that all requirements of the degree have been met.