simple question about self stabilizing robots

I'm a little confused on how self stabilizing robots work. Generally when I read descriptions on how they work they focus on the gyro and accelerometer, which I get. What I don't understand is how the body is connected to the wheels. In other words, if powered off, will the body just drop to the ground?

I'm assuming the answer is yes. Let's say that I'm correct, well then I have a followup question. Let's say that when powered off the weight is distributed such that the body will fall forward. If turned on, will the bot automatically move forward to redistribute the weight? I'm confused as to how this works; I guess I don't understand how moving forward pushes the body back if the body is loosely connected to the wheels. I imagine it works similar to a unicycle, but I'm still a bit confused. I hope I am clear with my question. If anyone can provide some feedback, I'd appreciate it.

Is there an Arduino question in there somewhere? Also, what kind of a robot is this, a ball, a humanoid, a two wheeled robot, something else? Never having made a self stabilizing robot I don't know. I would try another forum like Let's Make Robots for your question, or maybe Wikipedia.