What is PER?
Physics Education Research (hereafter known as PER) is a sub field of Physics (just like Plasma Physics or Gravitation Theory). We study how people learn physics, why they have difficulties, what happens on the way to becoming an expert, and how to teach physics better. We build (or borrow) on ideas from many fields, including education, cognitive science, learning sciences, and anthropology.
Do other fields have education research groups?
Sure, there is similar work going on in Chemistry, Biology, Engineering, and Math. People in Education Departments do work in all of these subjects, too, often in Science Education departments. Physics has been around longer (a mere thirty or forty years) than most, though, so we're probably the biggest.
Why do you need a PhD in Physics to do PER? It's not like your detailed understanding of electrodynamics (and I'm not claiming that I have such a detailed understanding) is going to help you figure out how people think about forces.
You don't need a PhD, it's true. Lots of people get a Bachelor's degree in physics and then get a PhD in Education and do really awesome work. But there are some advantages to the Physics PhD: (1) If you want to be a professor in a physics department, they'll expect your degree to be in Physics, not some other field. (2) Getting a PhD in Physics means that you become a part of the physics community, and not the education community. Thus you share (or at least understand) the ideas and practices that physicists value, so that you're teaching the things the community wants taught. Also, other physicists are more likely to take your ideas seriously. (3) One of my advisers makes a good argument that if we care about physics, then we have to care about how it's taught, because that's how the field grows. And if we care, we shouldn't leave the teaching to other people.
What do cameras have to do with physics?
You can study how people think about physics many ways: look at their homework, give them multiple choice surveys, interview them, and more. My group often learns about how people teach or learn by observing them as they are doing physics au naturelle - that is, we videotape them while they are in class or working on homework. Then we analyze the video, looking at who people talk to, the content of what they are saying, their gestures, their tone, etc.
I hope I have provided you with more that you ever want to know about the arcane subfield, which, for better or worse, is my life.