A pointed space $(X, x_0)$ is called an $H$-space if there is a pointed map $m:(X \times X, (x_0, x_0)) \rightarrow (X, x_0)$ such that each of the (necessarily pointed) maps $m(x_0, )$ and $m(, x_0)$ on $(X, x_0)$ is homotopic to $1_X \text{ rel} \{x_0\}$.
I don't understand anything in the bolded text. How are the maps $m(x_0, )$ and $m(,x_0)$ defined? Why are they necessarily pointed? What does it mean for the maps to be pointed on $(X, x_0)$?
I'm self learning this, so as clear an explanation as possible would be helpful.
The map $m(x_0,):(X,x_0)\to (X,x_0)$ is the map that sends $x\in X$ to $m(x_0,x)$. This is automatically a pointed map (that is, it sends $x_0$ to $x_0$) since $m$ is assumed to be a pointed map, so $m(x_0,x_0)=x_0$. Similarly, $m(,x_0)$ is the map that sends $x\in X$ to $m(x,x_0)$.