Let $\mathbb{H} = \{ z = x+iy \in \mathbb{C} | \Im(z)=y>0\}$ be the upper half plane. I often see that $\mathbb{H}$ is endowed with a metric which is written as
$$ ds^2 = \frac{dx^2+dy^2}{y^2}. $$

I do not understand what $s, d(\cdot)$ mean and how one can measure the distance of the points on $\mathbb{H}$ with that formula. And why do we have to take squares?

It seems like there is some background to that which I do not have. And I also do not know where to start. Could someone please explain me what all these things mean? Thank you!