What is Neural network, multilayer perceptron, activation function?

A neural network is a function built by composing parametrised linear maps with fixed nonlinear functions. A multilayer perceptron (MLP) is the fully connected case: $z_0 = x$, $z_l = \mathrm{act}(A_lz_{l-1} + c_l)$ for $l = 1,\dots,L$, and $f_\theta(x) = A_{L+1}z_L + c_{L+1}$, with parameters $\theta = (A_l, c_l)_l$. The activation function $\mathrm{act}$ is applied coordinate by coordinate; the usual one is $\mathrm{relu}(u) = \max(u, 0)$.