The difference between norm and modulus I'd like to know the difference between norm of a vector, ||v|| and the modulus of a vector, |v|
 A: I think the correct term when referring to general vectors is norm, indicated by $\| \|$. Modulus is the term specifically used for complex numbers (scalars), and reduces to the concept of absolute value when referring to real numbers. When viewing a complex number as a real pair in the complex plane, then modulus corresponds to the (euclidian) norm on $\mathbb{R}^2$. 
Note that modulus has different meanings in different mathematical contexts (eg Set modulus, congruence modulus, elliptic modulus, etc.)
A: This is an old post and Christiann already had a good explanation. But I just recently run into this confusion in my quantum assignment, so I thought to add this example, 
Given a complex wave function $\Psi(x,t)$, there are two quantities we can compute, the square norm $||\Psi(x,t)||^2$ and the square modulus $|\Psi|^2$(as called on wiki): 
\begin{equation}
||\Psi||^2=<\Psi,\Psi>=\int\Psi^*\Psi =\int|\Psi|^2
\end{equation}
The squared norm here is the probability of finding a particle in a certain region (the region you are integrating over) but the squared modulus here is the probability density.
See more in L-2 norm, Inner product.
