In California, employment is at will, but who has the authority to fire?
An officer of the Company? (I expect this is always true)
A direct supervisor (I expect this is also always true)
A board director?
An outside consultant hired by the company?
Company's part-time lawyer?

Anyone to whom the company gives that authority.

And Patty's answer is true in all 50 states. There is no state law and no Federal law that grants the right to fire only to specific positions. The janitor can fire if the company grants him that authority.

Then how does one know (in a particular situation) to whom such authority has been granted?

If you are walking down the hall and a secretary says "You're fired" do you just keep walking until you hit the exit? Or do you say something else? For example, "Show me a piece of paper that gives you that authority?"

Isn't that taking it to the extreme? If you're going to be fired, it's likely the secretary isn't going to do it. Your supervisor, manager, and/or Human Resources would be the logical people.

I definitely agree that a supervisor, manager or HR staffer is the likely person.

In the particular case I have in mind, the company did not use an employee to do the job, but an outside person. And that person's authority to fire was not clear.

I am still wondering what the right question to ask is in such a case.

Thank you! I appreciate the earlier answers.

In this case, were I the fired employee, I would call HR and play stupid. i.e. "I was wondering about my [name the benefit here]." or "what do you show as my termination date?" If the person doing the firing did not have the authority to do so, you would know at that point. But such authority does not have to be in writing, so asking the fire-er for proof is not worth asking.