Russell's Paradox is the statement "Does the set of all sets that don't contain themselves contain itself?" (Parse that as "The set of (all sets that don't contain themselves)".)
Now, obviously, the set of all sets contains itself, and in naïve set theory that isn't a problem: In naïve set theory, a set is just an unordered collection of any objects you can define; a set that contains itself poses no logical problems in an of itself.
However, once you define the set of "all sets that don't contain themselves", you run into an immediate problem: If it does not contain itself, it must, in which case it cannot, and so on, in a neat infinite recursion.
This is a real paradox, also known as a falsidical paradox, and a real paradox indicates a deep problem with the set of axioms it was derived from. (Banach-Tarski is not a real paradox by my definition; it is merely called a paradox because it is a counter-intuitive result.) Russell's Paradox blew naïve set theory out of the water; later set theories, such as Zermelo-Fraenkel (more commonly just called ZF), were very careful to not allow sets to contain themselves.
Now, obviously, the set of all sets contains itself, and in naïve set theory that isn't a problem: In naïve set theory, a set is just an unordered collection of any objects you can define; a set that contains itself poses no logical problems in an of itself.
However, once you define the set of "all sets that don't contain themselves", you run into an immediate problem: If it does not contain itself, it must, in which case it cannot, and so on, in a neat infinite recursion.
This is a real paradox, also known as a falsidical paradox, and a real paradox indicates a deep problem with the set of axioms it was derived from. (Banach-Tarski is not a real paradox by my definition; it is merely called a paradox because it is a counter-intuitive result.) Russell's Paradox blew naïve set theory out of the water; later set theories, such as Zermelo-Fraenkel (more commonly just called ZF), were very careful to not allow sets to contain themselves.