The standard deviation formula actually makes sense to me.
$\sqrt(\frac1N\sum(x-u)^2)$
However, I do not understand why in the standard deviation of a discrete random variable formula
$\sqrt( \sum(x-u)^2p)$
the $\frac1N$ does not appear. Why it does not? What could be a graphical interpretation to understand the formula?
If the discrete random variable is uniform over $N$ values, then the probability $p$ associated with each value $x$ that it takes is $1/N$, and you recover the standard deviation formula that makes sense to you.
More generally, random variables will not be uniform, so you replace $1/N$ with different probabilities $p$ associated with each $x$. This is the second expression in your post.