What does this (double absolute value like) notation mean?

Here,

$$\left\lVert\frac{\partial\bf x}{\partial s}\times\frac{\partial\bf x}{\partial t}\right\rVert$$

the inside will at last be a vector. and two absolute value signs have covered it. what does it mean?

Can someone explain it to me? $||\vec a||$

$\| a \|$ in general means the "norm" of $a$. Most commonly it means the Euclidean norm of the vector $a$. You could say "the geometric length of $a$" or "the magnitude of $a$" to refer to the same concept.

Be careful: there are many other norms that can be used to measure vectors, as well as norms that can be used to measure different sorts of objects entirely.

This generally indicate a norm in linear algebra and functional analysis. It can be thought as the length. For any $\mathbf{v} = [x,y,z]$ in $\mathbb{R}^3$, $\lVert \cdot \rVert$ represents the Euclidean norm $$ \lVert \mathbf{v} \rVert = \sqrt{x^2 + y^2 + z^2} $$