classical mechanics - What is the physical interpretation of the Poisson bracket

Apologies if this is a really basic question, but what is the physical interpretation of the Poisson bracket in classical mechanics? In particular, how should one interpret the relation between the canonical phase space coordinates,

$$\lbrace q^{i}, p_{j} \rbrace_{PB}~=~\delta^{i}_{j}$$ I understand that there is a 1-to-1 correspondence between these and the commutation relations in quantum mechanics in the classical limit, but in classical mechanics all observables, such as position and momentum commute, so I'm confused as to how to interpret the above relation?