What kind of hardware and software are required to build an AI system? What kind of hardware (processor, memory, graphics card, etc.) and software are required to build an AI system? How do we go about linking each and every one of them to their proper place and function?

What is needed besides the main hardware or software of a computer that will be used to make an AI system?
Many different components may go into building an AI system. AI is a broad topic encompassing, machine learning, statistical learning so an AI system could involve anything from a raspberry pi to a spoon. Your title refers specifically to processors of an AI system. The mathematics behind AI can be run on any processor. GPU, CPU, FPGA, and new Neural network chips. A neural network can be run on a Window PC a linux PC AI packages are not OS dependent. I can train a model on my laptop now. 

2019-11-14 20:10:57.503229:
Epoch 1/5
60000/60000 [==============================] - 8s 137us/sample - loss: 0.2925 - acc: 0.9141
Epoch 2/5
60000/60000 [==============================] - 8s 140us/sample - loss: 0.1426 - acc: 0.9574
Epoch 3/5
60000/60000 [==============================] - 9s 142us/sample - loss: 0.1059 - acc: 0.9674
Epoch 4/5
60000/60000 [==============================] - 8s 135us/sample - loss: 0.0862 - acc: 0.9732
Epoch 5/5
60000/60000 [==============================] - 7s 112us/sample - loss: 0.0744 - acc: 0.9763


Just like any computer application an AI needs a processor and memory to store information. Beyond that it is difficult what specifically you are looking for.

A configurable cloud-scale DNN processor for real-time AI