It depends a bit on the context, but the idea is that if two observables commute that means that measuring one will not affect measurement results of the other, and you can therefore think of the values of $A$ and $B$ in a given state as "simultaneously existing".
This means that you can characterize a state by the measurement outcomes you will get if you measure it in the different bases. For example, you might have a state $\lvert\psi\rangle$ which gives outcome "$1$" if you measure the observable $A$ and outcome "$-1$" if you measure the observable $B$, and therefore you can assign to the state the labels "$1_A,-1_B$", to represent this fact.
Note how this contrasts sharply with the case of noncommuting observables. If $A$ and $B$ do not commute, then measuring $A$ changes the results of measuring $B$, and vice versa. This means that you cannot meaningfully assign to a state labels representing what results you would get by measuring $A$ and $B$, because the order in which you do that changes the kind of results you would get.