Data demystified: neural networks -- how do they work?

Neural networks, deep learning, reinforcement learning -- all seem complicated, and the barrier to entry in understanding how these things work can seem too high. In this article, I'm going to explain the mechanics of a neural network in an intuitive way, using a worked example. Lots of explanations try to relate how a neuron in the brain works to an artificial neural network (ANN). However, unless you did biology, medicine, or neuroscience as a degree, you probably don't know how a neuron works in the brain, so it doesn't help you. I did medical degree with a specialism in neuroscience and I found the explanations of neurons'identifying straight lines, and loops' completely baffling, so don't feel disheartened.