What does it mean to vanish at infinity faster than any power of $|x|$? The Schwartz space is informally defined as the space of $\mathcal{C}^\infty$ functions which, together with their derivatives, vanish at infinity faster than any power of $|x|$.  I find this strange because no non-negative power of $|x|$ vanishes at infinity.  The explicit definition is 
$$\mathcal{S} = \{f \in \mathcal{C}^\infty : \sup_{x \in \Bbb R^n} (1+|x|)^N |\partial^\alpha f(x)| < \infty \text{ for all } N \in \Bbb N \text{ and all } \alpha \in \Bbb N^{n}\}.$$ Can someone please explain this?

It means that any power of $|x|$ will not grow fast enough when going to infinity to "counter" the speed at which $f$ and his derivatives vanish. 
So when you multiply $f \in \mathcal{S}$ or one of his derivatives by $|x|^N$ for any $N \in \mathbb{N}$, the limit will be $0$ when $|x|$ goes to infinity.
