In plain English
A neural network is a computing system made of interconnected layers of simple units, called nodes or neurons, loosely inspired by the brain. Each connection has a weight that is adjusted during training, so the network gradually learns to map inputs to useful outputs. Stacking many layers lets it capture increasingly complex patterns.
A worked example
A neural network trained on labelled photos learns, layer by layer, to turn raw pixels into a confident answer such as cat or dog.
Common confusion
The brain inspiration is loose and mostly metaphorical. Artificial neurons are simple maths, not biological cells, and the resemblance should not be pushed too far.