Russell's paradox, the set of all sets not containing themselves can be broken down to two statements:
A thing that contains all sets that don't contain themselves.
This thing/one such thing would necessarily qualify as a set.
Now, what was the definition of set that Russell went by that mandated the second statement?
Best Answer
The axiom that told Russell that he could consider that to be a set is called the comprehension axiom, which says that for any property $P$, there is a set $X_P$ such that $$ x\in X_P\iff P(x).$$ (We usually write $X_P$ using comprehension notation: $X_P=\{x:P(x)\}$).
The other important axiom here is extensionality, which says two sets are equal if and only if they have the exact same elements: this tells us that the above condition is a valid definition of $X_P.$
Russell applied the comprehension axiom to the property $P(x)=x\notin x$ and then derived his contradiction that $$ X_P\in X_P\iff X_P\notin X_P.$$
The lesson we learned from this is that the comprehension axiom is inconsistent. Thus we do not use it in axiomatic set theory. Instead we use weaker versions, most commonly the separation axiom of ZF that says for any set $A$ and property $P$ that $ \{x\in A:P(x)\}$ is a set.
The reason that Russell thought he could use the comprehension axiom is because it seemed naively true, had been used implicitly before in mathematics with no problems, and had even been singled out as a formal axiom by Frege. His discovery that this seemingly innocuous bit of mathematical reasoning led straightforwardly to an obvious contradiction led to a lot of worry about the formal foundations of mathematics in the ensuing decades, and many interesting things were discovered by logicians trying to interrogate exactly why comprehension failed and how to avoid similar problems.