In most of the production engines, the manifold is an assembly, which is often cast iron, that is designed to use the least metal, to make sure it is light weight and to occupy the smallest amount of space necessary to collect the exhaust gas from the multiple cylinders and combine those flows into one single pipe. Often the weight and space restrictions can result in a design that does not do the best job of venting the gases.
A header is another name for a manifold, but this is an enhanced manifold and the engineering parameters have been shifted from concentrating on the weight and the size, to concentrating on the optimal flow of the exhaust gases. In a set of tuned headers the pipe lengths are carefully calculated to enhance exhaust flow in a particular engine rpm (revolution per minute) range.
The header back is the portion of exhaust system from the outlet of the header to the final vent to open air. They are normally for vehicles without turbochargers. The turbo back is the portion of the exhaust system that goes from the outlet of a turbocharger to the final vent to open air. Turbo back systems are for vehicles with turbochargers. Some turbo back systems replace stock catalytic converters with others having less flow restriction.
Cat back is the portion of the exhaust system that goes from the outlet of the catalytic converter to the final vent to open air. This then will normally include the pipe from the converter to the muffler. The muffler is then the final length of pipe to open air.
Cat back exhaust systems are quite a popular performance enhancement. They usually use a larger diameter pipe than what they would on the stock system. The good systems will have mandrel bent turns. This allows the exhaust gas to leave with as little back pressure as possible.
The final bit of the exhaust pipe, which is the bit that is visible and where it vents to open air, is used to guide waste exhaust gases away from the controlled combustion in the engine. The exhaust pipe must be carefully designed to carry any toxic or noxious gases away from the vehicle. It often just ends with a straight or angled cut. Sometimes it may include some kind of a fancy tip. The tip will then often be chromed, and it more than likely will be of larger pipe than that of the rest of the exhaust system. This will then produce the final reduction in pressure, will also prevent the edges from rusting and can be used to make the appearance of the vehicle look better.