What is the use of the double modulus signs? So far as I can tell, the author uses $\|\cdot\|$ to mean the magnitude of a vector, but I have only seen the notation $|\cdot|$ to mean the magnitude of a vector. Is there any difference? If so, what? And if not, why bother?
 A: That's a common symbol for a norm in mathematics. I guess in your case it means the length of the vector. 
For more information about norms in general, see Normed vector space (Wikipedia).
A: The reason some authors choose to use $\| \cdot \|$ notation instead of $| \cdot |$ is to better distinguish between vectors and scalars.
For example, writing the identity $|kv|=|k||v|$ is somewhat ambiguous whereas $\|kv\| = |k|\|v\|$ is not.
A: In regular vectors, they're generally equivalent. (note: "regular" here isn't a special subset of vectors, it's referring to the common meaning of "vector")
However, there are vector spaces that aren't just a list of numbers in the way that regular vectors are. And in some of these vector spaces, the absolute value of something can be more vague.
For example, consider the vector space for functions that consists of cosine and sine functions. From this, we have
$$
\|\sin(x)\| = 1
$$
as $\sin(x)$ is a unit function in the space (due to normalisation). However, $|\sin(x)|$ represents an always-positive function, not the norm of the function.
When treating vector spaces formally, retaining the distinction is of value. When simply working with regular vectors, there's no special need, and it's the author's choice.
A: Basically what you are telling means a norm. So what is a norm, A norm is basically a function on a given vector space $V$ over the field $F$ of complex numbers. So a norm on $V$ is a function $p:V\rightarrow \Bbb{R}$ having the following properties:
1) $p(av)=|a|p(v)$
2) $p(u+v)\leq p(u)+p(v)$
3) If $p(v)=0$ $\iff$ $v$ is the zero vector.
Where $a \in F $ and $v,u \in V $
You can find other properties and other uses of norm here:http://en.wikipedia.org/wiki/Norm_%28mathematics%29
