Let us state axiom schemata without parameters, so as to facilitate readability, but with the understanding that we really mean with parameters.
Under this convention, the infamous Axiom Schema of Unrestricted Comprehension can be written follows.
$$\exists A\forall e(e \in A \leftrightarrow P(e))$$
Russell's paradox derives a contradiction from this schema. The proof is roughly as follows.
Substitute $P(e)$ with $e \notin e.$ $$\exists A\forall e(e \in A \leftrightarrow e \notin e)$$
Let $A$ be fixed but arbitrary satisfying the above sentence. $$\forall e(e \in A \leftrightarrow e \notin e)$$
Replace $e$ with $A$. $$A \in A \leftrightarrow A \notin A$$
Now presumably, when this paradox was first discovered, mathematicians tried very hard to salvage unrestricted comprehension. An obvious first attempt would be:
$$\exists A\forall e(e \in A \leftrightarrow e \neq A \wedge P(e))$$
This time, we obtain
$$A \in A \leftrightarrow A \neq A \wedge A \notin A$$
which isn't contradictory, indeed we may conclude $A \notin A.$
But, I'm guessing the modified schema falls prey to some kind of "modified Russell's paradox." Ideas, anyone?
Edit. As aws points out in the comments, this schema is actually consistent. Consider a model with a single element $*$ and define $* \notin *$.
Thus, we should at least assume there exists a non-empty set. Feel free to use something stronger, like the existence of an infinite set.