As someone who is an amateur designer but frequent purchaser of professional work I would like to offer an answer from the buyer's perspective. I say this with the utmost respect for the design profession and with sympathy to your present situation as I know it's painful to work without being paid.
I know I am putting my reputation points at risk with this answer. I am not trying to make people mad. I just want to offer a different perspective. I respect that many (most) of the people on this site will disagree. I hope your respect my right to disagree as well.
Analysis of the Situation
it is my understanding that designers don't work in PowerPoint
Graphic designers do work in Powerpoint and anywhere else good graphic design is needed (which is honestly everywhere). You seem to be under the impression that if a designer does not fit Adobe's narrow image of what tools and substrates should be used, that person is not a "graphic designer." That idea is simply wrong.
They needed some presentation decks, one of which needed to be editable (text to go on top of graphics) in PowerPoint. I told them that I do not work in PowerPoint as a graphic designer, what I can do is create images that can serve as a background template onto which text boxes on PowerPoint can be created.
The biggest problem I have with this is you're accepting work using a tool with which you have inadequate experience to fulfill the client’s needs. By doing so, you not only set yourself AND YOUR CLIENT up for failure (or an insane learning curve), but also prevent other designers who are better experienced and capable of fulfilling the client's needs from doing the work. (See Liability below.)
The buyer here seems to have clearly described their basic needs. You were not capable of meeting those needs due to your lack of experience in the work desired but offered an alternative. By offering this alternative after having heard the needs of your client, the client should feel safe assuming that as an expert you are offering them a true alternative, not a lesser or incomplete product. (Graphics that could eventually be used in Powerpoint slides is incomplete work based on the client’s needs.) They may have expected you to take an alternate route but for the end result to fit their required needs.
Think of what would happen if you went to a car dealer to buy a new car and they told you they have something just as good — all the parts unassembled. They promise you it’s essentially the same thing. Well, unless you know how to put cars together (I certainly don’t), it’s not worth anything to you as you needed a car. They give you the impression that it’s essentially the same thing but it’s not.
You should have declined the work at this point.
I initially did not want to invoice for this time, but they directed me to look into things after I told them I didn't work in PowerPoint.
If someone billed me for research that is obviously a prerequisite for the required task, I would have done the same. It's okay for a graphic designer to bill for research outside the basic task (like research on the automotive industry before doing an automotive design), but here it seems like something you must know before offering your services. Frankly, I would feel that the designer owed me for giving them the ability to learn basics on the job. They did not come to you saying "we need someone to research how to create designs for Powerpoint slides." They needed someone who knows how to do it. Whether you are paid hourly is irrelevant because your billable time is for design work, not basic research.
Perhaps a written contract that specifically states research outside the scope of Adobe tools is part of the work could have solved this but I don’t think it’s fair to assume they have to pay you to learn the basics of how to do what they want you to do.
They then sent me an email that my work was incomplete and unusable and would not pay me (p.s. they fired me before they even reviewed all of it, and what was reviewed was never stated to be incomplete or un-useable).
Unless your delivery included usable Powerpoint template slides (just like the ones that come with Powerpoint), then your work is incomplete. Whether or not they reviewed incomplete work is irrelevant (though it may have been rude that they didn't). What is relevant is whether the work is complete.
If they needed something to be created in PowerPoint from the start, they should have told me.
Other answers have a great discussion on contract law. These are great discussions and I don't want to take anything away from them, but unless you've got lots of money or a dusty law degree and plenty of time, you're better off just moving on. As is the client.
However (and I say this tongue-in-cheek), there are two sides to this situation and I'm not sure if you're entirely in the clear. If you wasted the buyer's time and gave the impression that you could meet their needs without being able to deliver you could be liable for the time and resources spent dealing with you. In some places, you could also be held responsible for paying the designer who does the work after you didn't deliver. If you were dealing with a very time-sensitive project worth millions of dollars, those lost millions could be your responsibility if you gave the impression they could count on you to do what they needed in the time it needed to be done.
Of course your description does not fit such an extreme example. I just wanted to make the point that contracts have two sides and liability can go both ways.
You put yourself in a bad situation. You should consider yourself fortunate that they cut you off before you sunk more resources into this. Don’t do work for which you are not prepared and do not expect the client to be satisfied with anything less than their desired work.