In some ways, it seems to me that the classical Poisson brackets are more mysterious than the quantum commutators. It's clear that various operators on wave functions don't commute, but the fact that the Poisson brackets are anti-symmetric is a little subtle.
In (one-dimensional) Hamiltonian dynamics, if you write the Hamiltonian as a function H(p,x) of momentum p and position x, then this gives rise to the equations of motion:
\frac{dx}{dt} = \frac{\partial H}{\partial p}
\frac{dp}{dt} = - \frac{\partial H}{\partial x}
That minus sign in the second equation is the source of the antisymmetry of the Poisson brackets. When you write H = K + V where K is the kinetic energy and V is the potential energy, then the minus sign is reflected in the fact that \frac{dp}{dt} = - \frac{\partial V}{\partial x}. Force is the negative of the derivative of the potential energy.
Anyway, with the minus sign in the equations of motion, you can write for any function f(p,x) of position and momentum:
\frac{df}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial p} \frac{dp}{dt}
= \frac{\partial f}{\partial x} \frac{\partial H}{\partial p} - \frac{\partial f}{\partial p} \frac{\partial H}{\partial x}
\equiv \{ f , H \} (the definition of the poisson bracket of f and H)
It's a little mysterious as to why that's an important concept in classical mechanics. But the most commonly used examples are:
\frac{d}{dt} f(x,p) = \{f, H \}
\{x, p \} = 1
Then the generalization to multiple dimensions is \{ A, B \} = \sum_j \frac{\partial A}{\partial x^j} \frac{\partial B}{\partial p^j} - \frac{\partial A}{\partial p^j} \frac{\partial B}{\partial x^j}, which leads to another famous example:
\{L_x, L_y\} = L_z (as well as cyclic permutations)
where L_x, L_y, L_z are components of the angular momentum.
