Square dancing is generally done in a square (ok, duh) of couples facing couples, so 8 people per square. The moves are very oriented towards the geography of the square.
Contra dancing tends to be done in longwise sets, in other words a long line of couples going down the dance hall from the top (where the musicians are) to the bottom of the hall. It's common in contra for every other couple to swap lead and follow (some think of men's and women's) position, so you have two long lines of alternating leads and follows facing their opposites. The couples closer to the top of the set progress 1 position towards the bottom each time through the dance, and the couples closer to the bottom progress 1 position towards the top each time through, and the dances repeat many times through, so most people end up dancing with most other people.
Does that explain it?
-Little Jack, twinkletoes
ps Now if only there was Morris Dancing on the Playa!!!!!!!