A good understanding of structuralism is something I think we are regaining in the United States, after our flings with certain "post-structuralist" modes of understanding things: there is a return to structuralism that is going on.
That is, unless it is a
discovery of structuralism, a return to something that was never really understood well in the first place.
I add this because if you look back at the old journals, you can't help but feel that there was a sort of odd reception of structuralism in the US, one that really missed a lot of its points. Mostly (I think) this was because of the work that people who had the French connections were trying to connect it to discourses that were active in the US that looked like structuralism (in the positive sense that we will develop below). So Geoffrey Hartman, for one (and he was not the only one to do this), would try and explain the structuralist project alongside Northrop Frye's work. This is a good start, but it betrays the fact that there was a certain hermeneutic tradition that was lacking in the US that perhaps was there in France, and allowed a purer understanding of the structuralist approach to take place. And then there was the approach, right after--or, more accurately, right in the middle of the structuralist reception--of certain "post-structuralist" discourses: this perhaps closed the era of structuralism in the US before it really could begin. It isn't an accident that Lacan, Derrida, and others introduced these new notions in a conference meant to consolidate the structuralist influence, in 1966 at Johns Hopkins--introducing and destroying it at the same time for people.
Before outlining what distinguishes this purer understanding, I'll name what I think is the tradition: Heideggerian phenomenology.
Some might be tempted to say: Saussure. But as important as Saussure was in France, Heidegger and phenomenology remain the wider background against which the key notion of structuralism can really be grasped. This notion, which will lead us to the distinguishing characteristic of a solid understanding of structure, is the
effectivity of the structure, the way the structure determines its elements.
Now, this effectivity is often explained still in Saussurian terms, around the discovery of the arbitrariness of the signifier: it has to do with the bonds forged horizontally, as it were, between the chain of signifiers, rather than vertically, or through reference to the signified. So where we have a group of signifiers, it is not the downward movement of each signifier's reference that matters,

but rather its ability to juxtapose itself beside other signifiers,

and bring about signification that way. This is a stronger claim, essentially, than "reference only occurs in a context." No: what this claims is that reference occurs only
through context, such that 1) the context is what "in the last instance" does the work of reference, not the signifier's relationship to the signified (i.e. the context takes over for the arbitrariness), and 2) the context itself becomes wider than a mere "context:" it is not reducible to anything like order, syntax, or exchange which may locally bring about the juxtaposition. The context becomes a structure, nothing that is still dependent upon the downward work.
So, what do we end up with? The effectivity of the structure on its elements is one that holds it together in the absence of any specific (i.e. non-arbitrary) determination of its elements. The structure's effectivity is what allows meaning (or rather, the function of meaning) to take place and be understood despite the ultimate arbitrariness of meaning.
But this is what I will call only a
positive definition of the effectivity involved. It is a rich, full concept of what it is for a structure to have the function of meaning. To this day, it is how we in the US are often taught the concept of structure.
What is more interesting, however, is the corresponding
negative phenomenon, that which the positive definition
implies. And phenomenology gives you a richer sense of this negative thing at work: it makes what is negative from the Saussurian perspective also positive. If the structure holds its elements together despite any local determination of those elements, what we have is a structure that
is or exists only insofar as its elements exist. In other words, the
limits of the structure are also defined here, such that we understand that while the structure is
transcendental (it governs all the elements despite their determination), it is also
finite (its governance does not bear upon anything other than what makes it up).
I will come back to this last word--finitude--in a moment, for it is precisely the distinguishing criterion of a purer understanding of structuralism that I am talking about. But it should already be clear from my vocabulary here (transcendental, finite) that these are phenomenological and specifically Heideggerian concerns. The effectivity we are talking about here is precisely the ontological one that Heidegger brings about in his laying out of Dasein: Dasein is made up of all sorts of specific elements, it is dispersed in these ways, but it holds itself together because the inner tendency (or necessity) of these elements is not to be without a determination (or rather, a determin
ing) that is larger than them, as it were--something that makes of them a whole, but which is not merely a summation of them or even of the nature of an organic, part-whole relationship. Heidegger precisely describes this as the transcend
ing (and not transcendental) essence of Dasein (or, more often, just Dasein's transcend
ence), which is quite accurate, because as we see, the ontological dimension which holds these elements together is only made up of these elements, and thus cannot be effective upon
everything,
universally: it is therefore a type of transcendence that goes against the very definition of the transcendental (it is finite), so another name must be sought out. Merleau-Ponty says that this transcendence or transcending is the "lining" of the elements like the lining of a jacket: it determines the contours of the elements but does not remain outside them, independently of them (he also calls it "the invisible"). Regardless, this transcending is the flip side, the negative of the determination of the arbitrary elements: it is the negative of a rich concept of structural signification which determines this signification as only existing insofar as its elements exist--and which is needed to bring this notion of signification really into its own.
Perhaps one of the best people to explain this is Louis Althusser, in
Reading Capital. One can see there that, liberated from the Saussurian framework we can still have a structuralist analysis, because this negative phenomenon is thoroughly grasped. In this case it is Marxism that is grasped structurally--despite the fact that we lack any reference to signifiers and signifieds, or arbitrariness more generally: the mode of production there is the global structure that determines all the rest of Marxist phenomena (culture, ideology, relations of production, etc.). But the important thing is the structure's remaining a process of transcend
ence and not a universal transcendent
al status: in other words, its effectivity will not be able to be separated from the phenomena that make it up: it will not be anything apart from these, its (here,
economic) expressions or effects:
The structure is not an essence outside the economic phenomena which comes and alters their aspect, forms and relations and which is effective on them as an absent cause, absent because it is outside them. The absence of the cause in the structure's "metonymic causality" on its effects is not the fault of the exteriorly of the structure with respect to the economic phenomena; on the contrary, it is the very form of the interiority of the structure, as a structure, in its effects. This implies therefore that the effects are not outside the structure, are not a preexisting object, element or space in which the structure arrives to imprint its mark: on the contrary, it implies that the structure is immanent in its effects, a cause immanent in its effects in the Spinozist sense of the term [that Althusser developed earlier], that the whole existence of the structure consists of its effects, in short that the structure, which is merely a specific combination of its peculiar elements, is nothing outside its effects.-"Marx's Immense Theoretical Revolution," 188-89 (I've removed Althusser's italics and added my own--he significantly adds a footnote to the phrase "metonymic causality," which attributes it to Jacques-Alain Miller, who tries to characterize the causality Lacan recognizes in Freud.)
The phenomena which the structure structures precisely make up the structure: their existence is the condition of the structure's existence.
As I said, this is essentially the
finitude of the transcendence of the structure that is involved. In Heidegger, the transcendence is precisely a function of the finitude of the phenomenon, as we saw but characterized only in a privative way by speaking of its non-universality. But more richly considered now, the this finitude is what makes determines the structure ultimately: the structure doesn't extend on and on in an infinite chain (the moving chain of signifiers in Lacan thus resists the structuralist mode of thinking the structure: the function of the Real is precisely developed to destroy the implications of this finitude), but remains a distinct entity which can be viewed, which can be seen existing, and which can therefore have limits insofar as the instances in which it makes its appearance or expresses itself are limited.
And it is this sense of finitude that is lacking in all the old talk about structuralism in the US. The idea came about that the structure lacked this finitude, and was therefore very ahistorical and all encompassing. And while this is true from a certain standpoint--insofar as the structure is considered finite and as something that exists--which is the standpoint precisely of the critiques of structuralism in France, in the US, without a sense of the finitude of the structure, this criticism seems to be more empty: it sounds like a criticism that can be levied against any method, which is that it pretends to a more universal or total set of ramifications, a universality or totality, than is always useful or safe. This also makes the post-structuralists just seem like a set of people reacting against structuralism (i.e., "post-structuralists," which is a uniquely Anglo-American name used to understand these people: they did not think of themselves as post-structuralists, as
after anything). In other words, it inscribes structuralism into a uniquely Anglo-American progressivist timeline, one that allows people to talk about it as if it were yet another event in the history of ideas--and not a set of theoretical and methodological propositions which need to be dismantled from the inside. This would also explain the odd staying-power of structuralism: it would have staying-power insofar as these labels and names, these horribly inadequate (and usually pretentious) ways of thinking of ideas, would not have actually touched that to which they claim to refer.
In the end, the more and more this aspect of finitude gets recognized and developed now in the US (which I think it is), the richer and richer the understanding of structuralism becomes.