What is the "power" a PhD advisor has over their advisees?

• In many programs, the advisor signs off on the PhD student's thesis. If the advisor doesn't approve the thesis, it is difficult to impossible for the student to graduate and earn the PhD.
• In many programs, the advisor is responsible for securing funding for their PhD students. If the advisor withdraws funding that the student was expecting, the student might not have another source of financial support.
• Advisors are responsible for training and mentoring their PhD students, which is a major part of the PhD. If an advisor withholds that training, their PhD student might not become a successful researcher without it.
• In general, advisors interact closely with their PhD students. If these interactions are unpleasant or abusive, the workplace becomes very, very uncomfortable.
• Faculty tend to have a lot more influence in an academic department than their PhD students do. An advisor can potentially make things difficult for their PhD student with other faculty in the department.
• In many cases, on applying for jobs after a PhD, a student is expected to have a recommendation from their PhD advisor. Without it, it can be much more difficult to find a job. Also see: How to handle not having my PhD advisor as a reference? and What to do when a thesis adviser refuses to recommend me?

So, I am asking what is the power an advisor has legally or illegally over their own advisees?

ff524's answer provides a lot of the positive impact an advisor has. I will elaborate a bit more on the... darker side of the advisor power relationship.

Because unfortunately, the most common situation for recognizing this imbalance is when a student has problems.

First, and most importantly, is the following: your advisor is your boss and ultimately who determines if you graduate (in concert with others, but normally your advisor has the significant impact). This is unique in the graduate world compared to normal jobs because oftentimes problems with your boss/advisor that in the working world would result in quitting and moving on cannot happen the same. A PhD student that is 2-3 years into their program is much prone to suffer through problems with an advisor than a person in a more normal workplace, because if they quit they generally will lose a fair bit of the last years. Whereas in the working world, if your boss is making life miserable it often is possible to go to a different company. You likely will even benefit career wise from this! But it's quite difficult to take 2-4 years worth of graduate work with 1 advisor and get another advisor to work with. This factor gets worse the further you are in your graduate study.

What this translates to is that as the PhD student gets further into their studies, the less they can "walk" and keep their academic efforts during their graduate work meaningful.

This doesn't generally cause problems because in theory, advisor and student work together well. But when it does, it makes PhD students feel completely powerless.

It really isn't that advisors have so much power, per se, but they effectively are a single-person gatekeeper to a person completing a multi-year project. Unfortunately, normally by the time you find out you have advisor problems you are often far enough along that you feel the need to finish anyways. And just slog through it.

Keep in mind your goals and your advisors goals may be different. For example, your goal of graduation may be counter your advisor's goal of finishing different research projects, publishing more papers, etc.

A few examples I know of firsthand from my graduate work and interactions with other graduate students:

• Your advisor is unresponsive/slow to get back to you. Your timeline matters more to you than to your advisor, generally. In cases where you need timely responses from your advisor (whether for research guidance, paper review, dissertation approval, whatever) you may be delayed for weeks or even months.
• Your advisor isn't engaged in your research. When you are first starting graduate work, having some direction and mentorship is fairly important. If you don't get this, you may flounder for months or years. When you recognize this, you likely are unable to get time with your advisor and be far enough in that you feel trapped.
• Expectations to spend too much time doing non-graduation relevant work. The reality is most graduate students spend some time on non-graduation specific work - whether teaching, working on research for grants tangentially (or unrelated) related to dissertation work, etc. Good advisors help keep your time spent on things relevant to your graduation. But if you are in your 4th year and your advisor "needs" you to spend 40 hours a week teaching a class...
• Unwilling to "approve" final defense without . Normally you need your advisor's buy-in or approval in order to make your final dissertation defense. An advisor expecting more work continually or otherwise not ever being satisfied can make this process hellish, particularly if it's combined with other factors here.

These are just a few potential problems that I have seen firsthand in different advisor/student relationships. I and others could list many more. Keep in mind that you are far more likely to hear of graduate school horror stories than "I had a totally normal PhD experience, my advisor was exactly what I needed, nothing more, nothing less" stories.

Notice that most of these are not really actionable short of working with your advisor, either. There is an entirely different set of issues that you can pursue more formal action against, but these are all "entirely permissible" problems that are nearly impossible to resolve should your advisor not desire that.

The non-actionableness makes it a double feeling of powerlessness. Not only can the student not really quit without sacrificing potentially years of work but there is nothing they can really do about it by staying, either. Combine this with entirely normal feelings of despair, meaninglessness, and futility that are present as part of graduate work? (I'm kidding about that. But only sort of,)

The key is to understand all the issues through the advisor/advisee relationship. When you do this, it becomes easy to see how a PhD student (and masters students, though to a lesser degree) can feel completely powerless.