Why at some airports/airfields you need to fly a non-standard circuit/pattern, what are the specific reasons for this and what are the differences between them while flying?
The specific reasons will likely vary with each individual airport, but the most common reason I've seen is noise abatement.
If the north/south runway is on the west side of town, using right traffic when landing on 18 will keep the planes from flying over the houses.
From the pilot's point of view you fly the circuit making right turns instead of left turns. It's easy to do and is covered in initial training.