Here are the people who do this for NASA planetary missions:
Here's a snapshot of object locations in the inner solar system:
http://ssd.jpl.nasa.gov/?ss_inner
There are a lot of challenging problem areas: rendezvous with small bodies (very irregular gravity fields), mission trajectory design, operations planning for flybys in the presence of, say, camera pointing constraints.
(@lutusp knows this already -- I'm putting it here just for interest.)
I can't speak to the linked simulator, but gravitational calculations are relatively simple -- they represent the solution to a numerical differential equation that, in small slices of time, solves:
f = G m1 m2 / r^2
Where:
f = force, Newtons
G = Universal gravitational constant: http://en.wikipedia.org/wiki/Gravitational_constant
m1 = mass of body 1, kilograms
m2 = mass of body 2, kilograms
r = distance between m1 and m2, meters
Here's my simulator, plus (further down the page) a complete description of the math. My simulator uses HTML5 and JavaScript:
http://arachnoid.com/orbital_dynamics
> I wasn't aware it was possible to run such a simulation on flash because each body results in 6x more calculation.
It's more accurate to say that the total computing time required changes as the square of the number of modeled bodies. That's worse than 6x, by the way, for a sufficient number of bodies. One solution is to use a large central mass and only compute the paths of some satellites with respect to the central mass, not with respect to each other.
The calculations that impress me (and I do a lot of this kind of math) are the cosmological modeling of galactic clusters of hundreds of thousands of stars, all interacting with each other, with no central mass to stabilize things. This requires a supercomputer and many hours of online time.
> Is this like plugging in the masses into a single equation?
No, because for more than two bodies, there's no closed-form solution (this is called the "three-body problem"). Solutions must be arrived at numerically. When NASA wants to compute where to land a spacecraft on Mars, they have to use numerical methods -- very good ones, but numerical methods.
http://en.wikipedia.org/wiki/Three-body_problem