Historical References:

"A History of Mechanics" by Rene Dugas is highly recommended reference, which includes discussion of Hamilton's work and its development. 

Dugas writes that Hamilton's geometric optics was a new method of formalizing the collection of results that had already been obtained, and capable of being interpreted in terms of wave propagation (in Huyghens' sense) and corpuscles (in the sense of the dynamical principle of least action). Hamilton says he was "struck by the imperfection of deductive mathematical optics", and wished to give the theory the same "beauty, power, and harmony" with which Lagrange had been able to endow mechanics. [Roughly quoting from Dugas, pp.390]. 

The principle of least action, as arising in the works of Fermat, Maupertuis, and criticized in Euler, Lagrange, was a great controversy. An important application was Hamilton's finding the proper *stationary* form, and Hamilton's principle of stationary action had less contentious metaphysical interpretations.

As to practical applications, Hamilton was an astronomer, and his first (1833) article on dynamics appears to be 
"On a General Method of Expressing the Paths of Light, & of the Planets, by the Coefficients of a Characteristic Function". [see google books preview]. In the "Second Essay on a General Method in Dynamics", Hamilton establishes the *canonical equations of motion* in terms of the partial derivatives of $H$. 

Jacobi appears to have simplified several redundancies in Hamilton's formalism, and is credited with *geometrising* the principle, in which case the characteristic function (Hamiltonian) has trajectories corresponding to the same *total* energy. 

Dugas writes that after Jacobi, further contributions were made by Liouville (1856), Lipschitz (1871), Thomson and Tait (1879), Levi-Civita (1896), and Darboux (last two chapters of his "Lecons sur la theorie generale des surfaces"). 

Levi-Civita's excellent text "The Absolute Differential Calculus"  contains discussion of geometric optics, Hamilton's principle, and describes how Einstein's general relativity attempts to generalize Hamilton's optics. See Part III, Chapter XI, pp. 287. 

I might even say the applications of Hamilton-Jacobi equations to classical mechanics continues even today(!) in the form of Monge-Kantorovich (Alexandrov-Brenier-McCann-etc.) optimal transportation. Also interesting is Nassif Ghoussoub's theory of self-dual variational problems, which is based on the classic correspondance of Lagrangian and Hamiltonian via the Legendre-Fenchel transform.