What is Commutator; compatible observables?

The commutator of two operators is $[\hat A, \hat B] = \hat A\hat B - \hat B\hat A$. The founding example, from $\hat p = -\iu\hbar\,\dd/\dd x$:

Two observables are compatible when $[\hat A, \hat B] = 0$: they then admit a common eigenbasis and can be known simultaneously; measuring one does not disturb a state sharp in the other. A set of commuting observables whose common eigenbasis is unique (a CSCO) is what “completely labelling a state” means — the labels $(n_1, n_2, n_3)$ of the box were exactly this.