The F53 chassis is a bare chassis used for class A motorhomes, most closely related to a medium duty truck chassis such as would be used for bread trucks or similar large step-van sorts of trucks. (There is a separate chassis Ford makes for them, the F59, but it and the F53 are generally very similar.)
The E350 and E450 are most commonly produced as cab and chassis or cutaway cab chassis, though there are bare chassis available as well. They're somewhat smaller and lighter duty than the F53/F59 chassis. They are used for class C motorhomes generally, and are based on the (now discontinued) E series vans.
The Transit chassis is another van chassis family, used for the current Ford vans, and is a little lighter duty than the E450 and, I think, E350 chassis. They are, however, quite a bit more modern in design and general comfort. These would be used for somewhat smaller class C's.
Other chassis found from time to time for class C's include the Chevy 3500 and 4500 (which is the GMC equivalent of the E350 and E450), the Sprinter (probably closest to the Transit), and for small units occasionally the Ram Promaster chassis (the smallest and lightest duty of the lot). All of these are van chassis. At the other end, various heavier truck chassis are occasionally found, generally in what are termed super-C motorhomes: the Ford F550 or F650, freightliner trucks, and probably a bunch of others. Many makers produce truck chassis.
The main differences between truck and van chassis have to do with where the engine, the seats, and the hood are in relation to each other. A truck generally has the engine in front, under the hood, and the cab behind it. (A cabover truck is, of course, a different story, but those are rarely seen in motorhomes.) A van chassis, on the other hand, typically has the engine roughly underneath the dashboard and the seats a little higher, with a short vestigial hood sufficient to service fluids and not a whole lot more. The floor of the cab in a van chassis is more or less flat, with the seats mounted on the floor, while a truck chassis tends to have a rather more lumpy floor.