A maxout unit can learn a piecewise linear, convex function with up to k pieces. 1
So when k is 2, you can implement the ReLU, absolute ReLU, leaky ReLU, etc., or it can learn to implement a new function. If k is let's say 10, you can even approximately learn the convex function.
When k is 2:
the Maxout neuron computes the function $\max(w_1^Tx+b_1, w_2^Tx + b_2)$. Both ReLU and Leaky ReLU are a special case of this form (for example, for ReLU we have $w_1, b_1 = 0$). The Maxout neuron therefore enjoys all the benefits of a ReLU unit (linear regime of operation, no saturation) and does not have its drawbacks (dying ReLU).
However, unlike the ReLU neurons it doubles the number of parameters for every single neuron, leading to a high total number of parameters. 2
You can read the details here:
1. DL book