A two-way cat reduces two pollutants, CO and HC. The catalysts (usually palladium or platinum) used have no effect on NOx. These are the cats used on most cars during the '60's, '70's and '80's.
NOx is generated solely due to the temperatures reached in the combustion process tearing molecules apart. There are two ways to reduce NOx output - generate less by lowering combustion temps, or clean it up after it is generated. The easiest way to reduce combustion temps with moderate effects upon mileage and power is to dilute the fuel/air mix by adding inert gases from the exhaust - thus, EGR (Exhaust Gas Recirculation).
Eventually, the boffins came up with a catalyst (usually Rhodium) that would reduce NOx and was economically viable. As a result, the cats normally used now have two beds (using these different catalysts), and since they reduce three pollutants, they are called three-way cats.
There are two types of three-way cats - the more common type on production cars uses air injection (from the air pump - but on earlier 928s, the air pump pushes air into the exhaust valve pockets to reduce HC and CO, not into the cats) into the cat between the first bed, which reduces (oxidizes) CO and HC, and the second bed, which reduces NOx. The heat from the first bed, along with the added air, makes the process pretty efficient. There are non-air injected three-way cats, which are cheaper and less effective, but work well enough to pass emission tests in most cases.
If you are adding aftermarket cats to your system, might as well use three-ways to reduce NOx - in some cases, that will allow you to meet more stringent testing without needing EGR.