the notched arc mated with the encoder wheel
It works the same as a mouse or trackball: the player moves the input device, which rotates the thingy, which is connected to a geared arc, which spins a notched wheel which passes through an infra-red sensing assembly... OK, it's less elegant and more complicated, but it works really well.
both wheels and their IR readers
That the N64 sticks tend to wear out is not a fault of the nifty optical encoding assembly, but rather a poor choice of materials for the stick itself. Were it not made of soft, unlubricated plastic it'd probably outlast most of the people playing with it.
one notched arc and the encoder wheels and PCB
The stick mechanism is a self-contained block, screwed shut to prevent the spring-loaded return-to-centre stick from exploding all over the place. It's full of little parts: The stick, two swinging armatures, a spring, a bowl/frame, two encoding wheels, a small PCB with the IR sensors, and a spring retainer. It's just mad, really: simple, but complex! Argh, whatever. Moving on!
a self contained unit
Ultimately it's a very clever solution, though I really cannot fathom why they went this route instead of using two potentiometres like the Playstation, Xbox, MegaDrive, Nintendo's own GameCube and Wii, and nearly every other 3D stick ever made as well. This device is durable, reliable and functional, but they're damned weird.
two notched arcs, one for each axis
Probably cheap though: except for the tiny PCB it's almost entirely made of plastic parts. A 'normal' potentiometer-based unit would be comprised of a lot of metal components... My guess is, despite this design's functionality, it was probably less durable than competing designs, and hence was never re-used.
All 3rd party N64 pads used normal potentiometre-based stick mechanisms.