The most important number is that for the GCWR (gross combined weight rating) as that is the maximum amount of weight that the tow vehicle can handle and includes everything (passengers, fuel, water, gear, trailer). It is based on the cabilities of the vehicle's engine, transmission and drive train, frame, suspension, wheels, tires, and other components.
The tow rating is how much the vehicle can pull depending upon whether it is a conventional trailer or a 5th-wheel trailer. Subract at least 1000 lbs. from this rating to determine the maximum dry weight for the trailer you plan to buy.
Payload is a fungible number as it includes an estimated number of passengers. If the truck has a crew cab and seats 6 people then the manufacturers calculate a passenger load of 900 lbs. and subtract that from the payload for the truck. It is why you will see the highest payload ratings for trucks with regular cabs.
The payload number provided by the manufacturer is calculated based on how the truck is equiped when it leaves the factory and takes into account the weakest link. Invariably the weakest link is the load capacity of the rear tires and these can be changed as can overload springs be added to the rear axle and heavier duty shocks and other suspension items like air bags and anti-sway bars). The rear axles on all the heavy duty (3/4 and 1-ton) trucks have a load rating of 8900 lbs. or more and so the payload is limited not by the rear axle and the wheel bearings but by the tires.
It is confusing but Ford and GM and Ram all provide towing guides for their vehicles and they are very well written and a good place to start. Regardless of what you buy and what package options it comes with from the factory you will need to add more items to get it ready for towing.