(DISCLAIMER) I am not an expert on this subject, and I only hope that my answer will attract better, and more complete ones.
The Meeus book that @Gerard Ashton mentioned is a great resource for amateurs, but I am worried whether it is accurate enough for, let's say, eclipse or transit predictions. Also I glanced at the site linked in the question and saw something like "A live view of Plutonian system". To be honest I don't think that the Meeus book can enable you to do that kind of thing.
As far as I know, serious calculations of planetary positions are done today with the help of dynamical ephemerides developed and maintained by specialized agencies, such as VSOP87 by the Bureau des Longitudes in France, and DE431 by Jet propulsion laboratory of NASA. The NASA Horizons system is an online interface from which you can get the planets' positions. (It's powered by some version of the DE ephemeris, I think.) It is well documented, so feel free to check out their documentation if you want to have a go.