I think you mean Hamiltonian operator gan and this is probably going to get way outside you understanding at a full level unless you have exceptional mathematics.
So lets se if I can simplify this ... I am going to take some liberties here so you can follow so this is a layman approximation.
Essentially the hamiltonian is going to going to be the sum of the kinetic and potential energies which is all reasonably easy in our normal classical world.
I have taken a liberty here in that we have to assume the system is closed and speeds and acceleration are not relativistic so we don't need to consider special relativy time effects.
So the Hamiltonian operator would describe the movement of energy across time.
All of your normal classical motion laws can be described by hamiltonian operators.
If you look at a normal orbital of a planet around the sun you are probably aware the planet goes fast near the sun and slows down as it goes away. There is kinetic energy and gravitational potential energy exchanging backwards and forwards and the planet has a total energy (see this example of the working
http://www.arachnoid.com/gravitation_equations/orbital_mechanics.html)
In that sample the combined energy I could rewrite the energy with respect to time of the planets position relative to the sun. Lets say I chose the position closest to the sun and called it time t = 0. I could then rewrite the energy with respect to that time and position as an equation and that would be a simple Hamiltonian operator.
Why are they useful? we already have the laws of motion and could calculate everything anyhow. That is true but say I gave you two orbiting planets at 90 degrees to each other, how do I calculate if and when the two planets will collide.
What the hamiltonian operator form of the orbitals will allowed you to do is solve for time the planets collide.
What happens if another sun drifts in and makes our sun bigger over a period of 20 years, okay it can't happen but you get the idea. What is happening is we need a way to change the energy in the system slowly over time so it correctly portraits the situation. Your classic laws assume things are fixed not evolving over time and at any point you calculate them they will be only correct for one point in time if the system is evolving.
In cosmology and QM hamiltonian operators describe how things partiles, planets, balls, objects or you and me whatever exchange energy over time.
If you want to know more it's probably best you do some reading because some of these details I find extremely difficult to simplify especially with English being my second well third language :-)
