What is the role of contact geometry in the hamiltonian mechanics?

I think the basic example is when you have a symplectic manifold $M$ with a Hamiltonian $H : M \to \mathbb{R}$. Then take a regular value $a$ of $H$, and look at the hypersurface $N := H^{-1}(a)$, which will be a smooth submanifold of $M$ of odd dimension. Then (probably with some more hypotheses that I forget now), $N$ will have a contact structure, and the corresponding Reeb vector field will agree with the Hamiltonian vector field $X_H$ corresponding to $H$. Recall that the value of the Hamiltonian function $H$ is constant along the flows of $X_H$. In terms of physics this is interpreted as conservation of energy or something like that. So in this basic example, contact geometry can be thought of as the study of Hamiltonian mechanics for a fixed value of energy.

In mechanics you often want to study systems whose Hamiltonian function depends on time (explicitly). For example, you can look at the motion of a charged particle in a time-dependent electric field. In such cases you are solving an ODE in the "extended phase space" ($\mathbb{R}^7$ in the above example), and not in $\mathbb{R}^6\simeq T^\vee \mathbb{R}^3$. Also, the translation between Hamiltonian and Lagrangian formulation of mechanics goes via Legendre transform, which fits very nicely in the framework of contact geometry. Contact geometry also enters mechanics through Hamilton-Jacobi theory and the "method of characteristics".

So you can think of contact geometry as the odd-dimensional ("non-stationary") analogue of symplectic geometry. You can go from one to the other by "symplectisation". For example, if you start with a manifold, $M$, then you have a tautological contact structure on $X=\mathbb{P}T^\vee_M$. The symplectisation of $X$ is $T^\vee _M-\{ 0 \}$, with the canonical symplectic form. Symplectisation maps contact diffeomorphisms to symplectomorphisms, etc. etc. In the opposite direction, if you are given a symplectic manifold$(M,\omega)$ with $[\omega]=0\in H^2(M,\mathbb{R})$, you can build a line bundle $E\to M$ with a contact structure.

Form the contact 1-form $\Theta = p \, dq -H \, dt$ on extended phase space $T^* Q \times {\mathbb R}$, the second factor being time and parameterized by $t$, the function $H = H(q,p,t)$ being the time-dependent energy or Hamiltonian, and $p \, dq$ denotes the usual canonical one-form on $T^* Q$ pulled back to extended phase space by the projection onto the first factor. (Assume $H \ne 0$ to get $\Theta$ contact.) Then the Reeb vector field for this 1-form, i.e the kernel of $d \Theta$, is the time-dependent Hamiltonian vector field, up to scale.

Arnol'd has a nice discussion of this in his Mathematical Methods in Classical Mechanics.