Why did mathematicians take Russell's paradox seriously? Though I've understood the logic behind's Russell's paradox for long enough, I have to admit I've never really understood why mathematicians and mathematical historians thought it so important.  Most of them mark its formulation as an epochal moment in mathematics, it seems to me.  To my uneducated self, Russell's paradox seems almost like a trick question, like a child asking what the largest organism on Earth is, and getting back an answer talking about a giant fungus.  

"Well, OK..."

The set of all sets admits all sorts of unmathematical objects right? Chickens, spoons, 2 $\times$ 4s, what have you...I can't imagine a situation where a mathematician solving a concrete problem would invoke a set like that.  The typical sets one sees in everyday mathematics are pretty well defined and limited in scope: $\mathbb{Z}$, $\mathbb{R}$, $\mathbb{C}$, $\mathbb{R}^n, \mathbb{Q}_p$, subsets of those things.  They are all built off each other in a nice way.

But clearly important people think the result significant.  It prompted Gottlob Frege, a smart and accomplished man, who Wikipedia tells me extended the study of logic in non-trivial ways, to say that the "foundations of [his] edifice were shaken."  So I suppose Russell's paradox is "deep" somehow?  Is there a practical example in mathematics that showcases this, where one might be led horribly astray if one doesn't place certain restrictions on sets, in say, dealing with PDEs, elliptic curves, functional equations?  Why isn't Russell's paradox just a "gotcha"?
Russell's paradox means that you can't just take some formula and talk about all the sets which satisfy the formula.

This is a big deal. It means that some collections cannot be sets, which in the universe of set theory means these collections are not elements of the universe.

Russell's paradox is a form of diagonalization. Namely, we create some diagonal function in order to show some property. The most well-known proofs to feature diagonalization are the fact that the universe of Set Theory is not a set (Cantor's paradox), and Cantor's theorem about the cardinality of the power set.

The point, eventually, is that some collections are not sets. This is a big deal when your universe is made out of sets, but I think that one really has to study some set theory in order to truly understand why this is a problem.
Fortunately, set theory isn't that important to modern math.  But category theory is, and an analogue of Russell's paradox stops you from constructing what would otherwise be one of the most obvious things to construct, the category of categories.

In practice, I've seen a bunch of expositions that go "Let $\mathbf{Cat}$ be the category of all (small) categories..." with a bit of a wink and a nudge to get you on the right track.  So I don't think this is a terribly important issue -- but it does give a natural place for a working mathematician to get accidentally involved in set theory.

By Russell's paradox proper, there's also no set of groups, rings, topological spaces, and so on, but this can be remedied by introducing proper classes, and as far as I can tell, you'd never want to topologize the class of topological spaces or something like that.

Edit: I realized your question was more historical.  Russell came at a time of tremendous vagueness in math, and his work was largely an effort to eliminate such vagueness.  It was surprising that such a project could possibly have nontrivial results.  Frege, specifically, had been working on a massive treatise on logic that was totally destroyed by Russell's paradox, because it relied so heavily on sets whose elements could come from anywhere.  If you're interested, Logicomix is a really good graphic novel about Russell's life and work -- addressed to non-mathematicians, but a good read and helps motivate issues like these.