Russell's Paradox is a paradox in set theory. One can classify sets into one of two categories. The first category contains sets that are not members of themselves. This contains most of the sets that we run into in everyday life. For example, the set of all penguins falls in this category, because the set of all penguins is a set, not a penguin. The second category contains sets that are members of themselves. The set of all non-penguins, for example, is not a penguin and thus is a member of itself. So is the set of all sets.
Now, take the set of all sets that are not members of themselves. In which category does it belong? If this set is not a member of itself, then it is a member of itself. If it is, then it isn't. So, this set is a member of itself if and only if it is not a member of itself, which is the paradox.
This paradox was very significant when it was discovered by Bertrand Russell in 1902. Around this time, attempts were being made to put mathematics on a foundation of set theory. However, if this foundation contains a contradiction, this is a very big problem, since, according to the laws of logic, one could then prove any mathematical statement to be true.
Obviously this paradox needed to be resolved. The "naive set theory" of the time allowed for a lot of leeway in how a set could be defined. The way that Russell and others resolved the paradox was to disallow predicates such as "the set of all sets that do not contain themselves" to be valid definitions of a set.
This paradox is similar to the Cretan Liar paradox.
View An article about Russell's Paradox at the Stanford Encyclopedia of Philosophy.