The Other (the constitutive other, they sometimes say) is the not-you that allows you to think of yourself. Like the adage that evil is necessary so that we can understand and appreciate good, so to does the self need an Other to understand itself. This, presumably, happens first in interpersonal relationships. I know I'm me because I learned as an infant that my body ends here and my mother starts there. (My understanding of it is inflected by phsychoanalysis for reasons that will manifest later on, but it is worth noting that it the Other does not originate in Lacan--and is not explicitly named in Freud, as far as I've heard--but shows up in Derrida, who, if I might indulge in extrapolating from the coupla lines I just read on Wiki, probably borrowed it from Heidegger.) The constitutive other plays an important role at other levels as well. I know I am a nerd because I'm not like that ass with the backwards hat over there. I know I'm white because I've seen a black person. The implications of the concept of the Other on politics and identity are significant and quite serious.
Spivak (though she is too graceful to use the word) points at Kant as an Enlightenment-era origin of self-knowledge through difference--that is, the politics-of-the-same that privilege that which belongs by identifying that which does not. Kant's Critique of Judgment constitutes the Educated Man--the privileged Judge who can competently say what is and is not beautiful, who can experience the sublime--by differentiating Him from the Feueländer and Neu Holländer (the colonized Other). William Haver, who prefers the term "abject," identifies the need for a constitutive other as one of the central problems of Humanism. Deleuze and Guattari strive mightily to disperse the subject position, arguing that the self is not coherent, that I am a we who should embrace and explore our schizophrenia.
So part of my discomfort, I would say, is that McClary, and others (teehee) are unclear about which of these positions they are adopting, or which new one they are establishing. The term Other is slippery. I can look at you and say you're my Other, but what politics does that bring to bear? (two bears too bare...)
There are two reasons we need to be careful with a word like this.
Derrida doesn't call it the Other either, I don't think. He has his Parergon, his supplement borrowed from Kant (I don't recall if Spivak dips into the Critique of Pure [?] Reason for her reading), that is never part of the equation but is always part of the equation. Every picture needs its frame, but no frame is part of the work of art. Adam needs Eve, but only so he can make more boys. Is this the Other? But the supplement is also part of. Carmen is certainly part of her opera, and part of Spain. But this is obviously quite different from the psychoanalytic trace, which needs complete separation. Sure, our mothers are important to our lives, but they are necessarily outside our limits of corporeality, or else there is not sense of self: the other would not then constitute. Further, Carmen is not an excess in Carmen. She is the focal point, even if not the protagonist (I agree with McClary in assigning this role to Don José). The only sensible reading of McClary (and I think this is what she meant) is that Carmen is a cultural other in Spain, and that Bizet's opera demonstrates that the Other who transgresses, rather than grinning and bearing it, is outside the law. Outside the law is not a happy place. You get stabbed there :(
I think the problem at this point is one of verbal specificity. What exactly does she mean? McClary asks at one point if Bizet's subaltern can speak. But Carmen is never a subaltern (you can tell because she's speaking). When McClary notes that Carmen of course is only Bizet's voice, she is also arriving at this conclusion. Then can Carmen be the Other? The politics of representation seem to make this word too slippery.
Irigaray would be useful (problem 2, I guess, if I'm sticking to an ill-conceived list). She suggests that the constitutive other (and she's mainly looking at Hegel and Freud, so the concept is slightly different) is not only essential for the grounding of the metaphysical subject (which is "always masculine") but is always specular. That is, the other--that guy, or Woman, or the racial other...whatever--is always merely a reflection of the self. Essentially, she is accusing the history of western thought of being always only so much solipsism. She is doing Schopenhauer one better. For him, there is only the self and the rest of the world: I cannot understand you in terms any more elevated than I understand that jar of peanuts over there. For her, there is only the self. Or rather, western metaphysics grants access only to the self. And of course, this same metaphysics coerces women into denying "the specificity of their relationship to the imaginary" (I'd generalize this further, as D+G seem to do: the theory of the coherent subject effaces the possibility of freedom).
I'm tired and the point, I think, is mostly made. Essentially, the constitutive other is too diverse a subject (teehee) to make the term useful without extended and rigorous discussion, for which there is little room in any text that wants to discuss anything further.