It's difficult to say exactly what personality traits are characteristic of a color/breed - they're all very different. However, in my experience, I would agree with above, that they have a tendency to be very, very stubborn. The appaloosa I've had have all been very rational, fairly bombproof and smart, but have had a little bit of an evil side - they did what the wanted when the wanted.
I had one that was very time consuming to train, and very lazy. Another that was very quick, extremely athletic, and while he was honest, he did what he wanted. For example, he would jump in a 4' line intended to be 6 strides, make it an easy 5, and there was no stopping him. Nothing crazy, he just wasn't going to consider adjusting.
My first training project was an appy, and if nothing else, I learned patience.