Glossary
Augsburg Chess is a fairy condition in which it is legal to move a unit to a square already occupied by a unit of the same color.
This results in a combined unit which shares the power of the two units. This combined unit can be moved "normally". It can also be split back into two components when you move one component and let the other one still on the from-square. In the end, every unit is a summation of several individual "atomic" units.
Beyond these general principles, the first point you need to understand is that, in Augsburg chess, "Queens" do not exist any more. They are Rook + Bishop combinations. So that a game may start 1. e2-e4 d7-d5 2. Bd1-h5 Rd8-d6 leaving a white Rd1 and a black Bd8!
Other, less important, details are:
I don't have a nice pedagogical example of how Augsburg chess gives rise to fun retros (send me one). There exist some variants of these rules but most problems (including number 6556 from feenschach, Nov 1993, the real motivation for this page) can be solved without this knowledge. I'll update this page when I can find some more information.
(In the meantime, thanks to Bernd Schwarzkopf for posting the above rules, and thanks to Juraj Lörinc for answering some questions)