Despite the biological name, an artificial 'neuron' is a small, precise piece of math: take some numeric inputs, multiply each by a learned weight, add them up plus a bias, then squash the result through a nonlinear function. That's it. A 'neural network' is just many of these stacked and connected in layers.
Why stack layers at all?
Input layer
raw features: pixels, tokens, numbers
Hidden layer(s)
learned intermediate representations
Output layer
prediction: class, next token, price
A single layer can only draw straight lines through data (linear separation). Real-world problems — telling cats from dogs, predicting the next word — need curved, twisted decision boundaries. Stacking layers with nonlinearities in between lets the network compose simple pieces into arbitrarily complex shapes.