What does "Machine Learning" actually mean? What's going on inside the blackbox?
The graphics below shows its learning process. You, programmer, define the network structure and your computer fine-tunes the sensitivity of interconnecting synapses, step by step, with the aim of reducing prediction error (cost function in their jargon).
Artificial neural networks mimic biological neural networks (perfect tautology!). Hidden nodes swallow all inputs (as polynomial) from the left side, digest them into [0,1] value (as activation function) and vomit it from the right side to the next nodes. As activation functions are not linear, you can not solve this network analytically.
Computer scientists try to solve the network, in a sense of finding out the better widths(parameters) of interconnecting lines, by repeating slight change of parameters and whole cost(error) check. This process sometimes succeeds, sometimes fails. You can see the trial-and-error process in the ever-fluctuating image below.
The more hidden nodes, the more the network will learn. This version with 7 hidden nodes will try to understand, after several thousands iterations, the leftmost red point in the crowd of blue points. (Don't try this with mobile browsers)
Denny Britz writes here about the structure of a simple neural network and its algorithm.
Sorry for my poor English, if you call it English.
data: scikit-learn(sklearn.datasets.make_moons)