A regular point $x_0$ of a function $y=f(x)$ is a point at which at least one partial derivative $\partial f/\partial x_i$ is not zero. A regular value $y_0$ of a function $y=f(x)$ is a point so that every point $x_0$ at which $f(x_0)$ is equal to $y_0$ is a regular point. By a theorem of Sard, almost every value of a smooth function is a regular value.

Take a Hamiltonian function $H$ on a symplectic manifold $X$, i.e. a Hamiltonian system with Hamiltonian $H$. If we pick a regular value $J_0$ of $J$, then the level set, i.e. the set of points where $J=J_0$ is a submanifold of $X$ invariant under the Hamiltonian flow of $J$. Suppose that the set of flow lines are parameterized by a smooth manifold $Y$ of dimension one less than the dimension of the level set. Let $\varphi\colon X \to Y$ be the map taking each point $x\in X$ to the flow line of $J$ through that point $x$. Then there is a function $h$ on $Y$, so that $H(x)=h(\varphi(x))$ for any point $x\in X$. (We say that $H$ descends down to $Y$, and write $h$ as $H$.) This $h$ is a Hamiltonian of a Hamiltonian system on that smooth manifold $Y$, for a natural symplectic structure. Simplest example: if $J=p_n$ in Darboux coordinates, i.e. $H$ is independent of $q_n$, then we can use coordinates $q_1,\dots,q_{n-1},p_1,\dots,p_{n-1}$ for that quotient manifold. In that case, we can write $H(p_1,\dots,p_n,q_1,\dots,q_n)$ as a function $h(p_1,\dots,p_n,q_1,\dots,q_n)$. (I am hoping that you are familiar with these $p$ and $q$ Darboux coordinates.)

A symplectic structure on $Y$ means that there is some way to set up a Hamiltonian system on $Y$, but a precise definition requires a familiarity with differential forms or some other mathematical structure which I can't perhaps give you. The point is then that $\varphi\colon X \to Y$ takes Hamiltonian paths of $H$ on $X$ to Hamiltonian paths of $h$ on $Y$, for the associated Hamiltonian system with $h$ as Hamiltonian function.

It is true, as Michael says, that this essentially says that ignorable things are ignorable, i.e. that if you don't use $q_i$, you can skip using $p_i$ too. If you would like an easy way to see that, note that near any regular point of a function $J$, there are Darboux coordinates in which $J=p_1$. So that reduce the local picture to the study of Hamiltonians $H$ for which $\{p_1,H\}=0$, and this you can work out trivially by hand.

Looking back at this answer, it doesn't belong on mathoverflow, and I don't think the question does either.