A neural network is a computation model built from layers of simple units that transform inputs into outputs through learned numeric weights. Loosely inspired by neurons, but really just very large, trainable math.
Neural networks are the substrate of deep learning. The term tells you how a system works mechanically — layers and weights — without implying anything brain-like about its 'understanding'.
In an image network, early layers respond to edges and colors, middle layers to shapes, and late layers to whole objects like faces or cars.