Depth means layers of increasingly abstract features
A neural network is called 'deep' when it has multiple hidden layers stacked between the input and output, as opposed to a shallow network with just one. Each layer transforms the output of the layer before it, and this stacking lets a deep network build up increasingly abstract representations of the input. In an image classifier, for example, early layers might learn to detect simple edges and color gradients, middle layers combine those edges into shapes like curves and corners, and later layers combine those shapes into recognizable parts like an eye or a wheel, until the final layers recognize whole objects like a face or a car.
This hierarchical structure roughly mirrors how biological vision seems to work too, with simple feature detectors feeding into progressively more complex ones. It's also why deep learning tends to outperform shallow models on complex, high-dimensional data like images, audio, and text, the depth gives the network room to build a layered understanding rather than trying to map raw pixels directly to a label in one giant leap.