What is the physical differences between the different classes of hitch?

I think it's related to the strength/size of the steel more than anything. The higher the class, the thicker the material used and possibly the number of points at which it attaches to the frame. Also, the higher the rating, the further back the attachment points will be to get closer to being over the tow vehicle's axle for weight distribution.

From the looks of this chart compact cars seem to be limited to a class 1 hitch, class 2 for midsize and trucks from there on. There are some sites on the web that provide the components and you weld it together yourself, or pop into a hitch store and ask. They're on every other corner in Texas, don't know about your neighborhood, but avoid U-Haul, too pricey and they generally don't know more than the consumer about the hitches.