The first cause argument asserts that everything we see around us is causal in nature, therefore if one goes up the chain of causality, one will encounter the First Cause, and that would be God.
Russell criticizes this argument by quoting a story from his days of youth.
I may say that when I was a young man ... , I read John Stuart Mill's Autobiography, and I there found this sentence: 'My father taught me that the question, ``Who made me?'' cannot be answered, since it immediately suggests the further question, ``Who made God?'' ' That very simple sentence showed me, as I still think, the fallacy in the argument of the First Cause. If every thing must have a cause, then God must have a cause.
He goes on to say,
There is no reason why the world could not have come into being without a cause; nor, on the other hand, is there any reason why it should not have always existed. There is no reason to suppose that the world had a beginning at all.
There are two criticisms that can be made on his argument. First, there's no reason to assume that if everything has a cause then God must have a cause. This argument is similar to saying that if objects A and B possess the property P, then an object C also possesses the same property. Second, when one surmises that the world had no beginning, it does not constitute a demonstration of the validity of the hypothesis. Whether the world had a beginning at all or not has to be decided through other channels. Scientific study indicates, contrary to the hypothesis, that the universe did have a beginning.
Russell's attack on the First Cause Argument appears to be inconclusive at best.