Dan,this is fairly simply. For example, if you are playing a song in C, the basic chords are C - G (G7) -F. If you play that same song in the key of D, you are moving everything one step/key, thus you would end up with D (from C) - A/A7 (from G/G7) - G (from F).
In other words, since you are simply moving one step from the key of C to the key of D, the corresponding chords move one step as well (of course, there are only 7 basic chords). This theory works the same with minors, as shown in your "Green Fields" chords. For example, if you were moving from the key of C to D, then your minors would move one step/chord as well (Am would become Bm and Dm would become Em). I'm sure you can see how the chords would move, if you you moved from the key of C to the key of E, for example. Everything would move 2 steps.
I think this is essentially correct. There are others on the Mudcat who may be able to give you a more detailed music "theory." Regards, Jim