Basically as I understand it, there are the chip designers, like AMD, NVIDIA, Intel, and so forth who do all the simulation, testing and design for the various chips.
But the actual literal manufacturing of these chips in bulk is done by "chip foundries", which are essentially contract factories that do nothing but manufacture microchips. Foundries have various capabilities- 3 nm process, 5 nm process, etc... which limits what sorts of chips they can produce- for example, most modern-day high performance PC chips are 7 nm, but some high-end phone chips are starting to be made using the 5 nm process. Basically the smaller the process, the more components (i.e. transistors) can be crammed onto the same area of silicon wafer, making for a more capable chip.
From what I gather the shortage is due to a perfect storm of sorts of weirdness in ordering by electronics manufacturers, combined with dramatically different demand and demand patterns and just-in-time manufacturing patterns, so that while the foundries are going full-tilt, the combination of shifted orders and greater demand has meant that they can't produce enough of anything to satisfy demand. It's just going to take some time for them to fulfill orders and pivot to others and slowly satisfy the demand over time.