The activation function in a neural network models the way a single neuron responds to different levels of input activation. It is intended to emulate the way that neurones in the brain have a non-linear response to stimulation from other neurones. The simplest activation function, used in early perceptrons is a threshold or step function, but most neural networks use some form of sigmoid activation function, which is like a smoothed step. The continuity of the sigmoid makes it easier to train, in particular enabling backpropagation.
Used on page 146
Simple step/threshold activation function
Logistic curve, a common sigmoid activation function