I'm hoping someone can educate me with regard to the differences between an AOW and an SBR?

Can someone cite their respective defintions and transfer taxes and offer examples of each?

I'm primarily interested in building a 10.5 or 11.5 inch barrel semi-auto AR15. I'm guessing this would fall in the SBR category? I've owned class 3 FA weapons back in the day so I'm somewhat famliar with the application process. In the case of an SBR, would a specific lower receiver have to be designated as the SBR or once the application is approved can I use the short barrel upper on any lower?

Thanks.