What is Commutator?

Example 1.6 (The canonical commutator)

For any function $f$, $\hat x\hat pf = -\iu\hbar xf'$ and $\hat p\hat xf = -\iu\hbar(f + xf')$. The difference is $[\hat x,\hat p]f = \iu\hbar f$, so $[\hat x,\hat p] = \iu\hbar$: position and momentum do not commute. This single relation underlies the uncertainty principle and, below, the whole spectrum of the harmonic oscillator.