Neural network is a fairly broad term that covers a number of related methods, but in our case, we will focus on a feed forward network that trains with backpropagation. I'm not going to waste our time discussing how the machine learning methodology is similar or dissimilar to how a biological brain works. We only need to start with a working definition of what a neural network is. I think the Wikipedia entry is a good start.

In machine learning and cognitive science, Artificial neural networks (ANNs) are a family of statistical learning models inspired by biological neural networks (the central nervous systems of animals, in particular, the brain) and are used to estimate or approximate functions that can depend on a large number of inputs and are generally unknown. https://en.wikipedia.org/wiki/Artificial_neural_network

The motivation or benefit of ANNs is that they allow the modeling of highly complex relationships between inputs/features and response variable...