Does the set of all those sets that do not contain themselves, contain itself?
This is known as the Russel's Paradox
Can you explain that for a layperson?
Essentially the paradox is exactly as OP said-- first you a define a set R such that it includes all sets that do not include themselves. (lets call them X sets). If R is not a member of itself, then by definition it would be an X set. However, if it is an X set, then R does include it, and boom, paradox!
Several systems have come to work around this paradox, generally by reducing the strength of statements (in Zermelo's Axiomatic system you simply cannot have a set like R). Did that make sense, or did I just make it worse?
You just made my head hurt. Please don't do that again!