Question about absolute value In $\mathbb{R}$, I know that 
\begin{equation*}
|x|=
\begin{cases}
x&\mbox{$x\geq0$}\\
-x&\mbox{$x<0$}
\end{cases}
\end{equation*}
What's the $|\cdot|$ in $\mathbb{R}^d$? Is it $|x|=(|x_1|,\ldots,|x_d|)$?
 A: The notation $|x|$ is really asking for the size of $x$, i.e. how far away it is from zero. Hence, both $+2$ and $-2$ are two units away from zero and so we have $|-2|=|+2|=2$.
In two dimensions we can ask about the "size" of the point $(x,y)$. We usually write this $\|(x,y)\|$. If you use Pythagoras' Theorem, you will see that the point $(x,y)$ is a distance $\sqrt{x^2+y^2}$ from the origin $(0,0)$. You can think of this as the "size" of $(x,y)$.
The same idea extends into three, four, and even higher dimensions. In three dimensions, the point $(x,y,z)$ is a distance $\sqrt{x^2+y^2+z^2}$ from the origin $(0,0,0)$.
In $\mathbb{R}^d$, the point $(x_1,x_2,\ldots,x_d)$ is a distance $\sqrt{x_1^2+x_2^2+\cdots+x_d^2}$ from the origin $(0,0,\ldots,0)$.
A: This really depends on the context. For example, $\mathbb R^d$ with pointwise ordering could be considered as a Riesz space (vector lattice).
In such spaces the absolute value is defined as $|x|=x^-x^-$, where $x^+=x\vee 0$ and $x^-=(-x)\vee 0$. 
So in $\mathbb R^d$ with partial ordering $$(x_1,\dots,x_n)\le (x_1',\dots,x_n') \qquad \Leftrightarrow \qquad x_1\le x_1' \land \dots \land x_n\le x_n'$$ you indeed get $$|x|=(|x_1|,\dots,|x_n|).$$
A: There are multiple meaning for $||x||$ or $|x|$when $x\in \mathbb{R}^d$. Generally $||x||$ is called the norm of $x$. 
The norm is defined only by certain characteristics. Any function that follows these characteristics can be considered a norm. So based on the context one can choose the norm one wants. The norm given in Fly by Night's answer is called the Euclidean norm and commonly written with single lines as you have $|x|$ when $x\in \mathbb{R}$ (Can use single line in higher spaces too, a matter of preference). Other norms in Euclidean space include Manhattan norm, p-norm etc. The characteristics of a norm function are: absolute scalability, follow triangle inequality and zero only at zero. From these non-negativity follows.
Please refer http://www-solar.mcs.st-andrews.ac.uk/~clare/Lectures/num-analysis/Numan_chap1.pdf and http://mathworld.wolfram.com/VectorNorm.html for a quick reference.
And refer Symbol for Euclidean norm (Euclidean distance) for a discussion on notation.
