This is truly just the beginning of some thoughts. My colleagues and I think and speak about reduction all of the time-- for us, it is one of the most important questions in winemaking. We are so much at the beginning of our thinking, so these remarks are just a first sketch of some thoughts that are still provisional.
What I mean by reduction: the state in the wine (or juice) of certain molecules that would very much like to bond with oxygen but find themselves unable to do so. Very often, the molecules that we have in mind are species and variations of hydrogen sulfide. I am not even sure whether other species of molecules enter into the effects that we are thinking about here.
How we bring about reduction: we find that if we avoid racking wines and extend lees contact, we can bring them into states of reduction during their maturation. We find that if we do not add nutrients (or do not add the full amount of recommended doses of nutrients) to fermentations, we can achieve reduction at very early stages of the wine's life. We find that this kind of reduction rarely persists in a negative form-- unless a nutrient disaster occurs.
We encourage or pursue reduction in certain wines (LSB, Babylon, Riquewihrt) to different degrees and with different end in mind. In general, we are looking for stonies or salinity, looking to underline or counterpoise fruit with something opposite-- but not dirty. We do not want reduction to go that far. We find that almost always the reduction that we sense early (say in fermentation) transforms over time into something more complex and less easy to identify-- a seamless part of the wine, rather than a component of it.
Does reduction "cause" "minerality"? The difficulty of the question is pointed to by the quotation marks. Let me say only this: it has been my experience, since first discussing the question with Jason Berthold in 2005 and 2006, in some or perhaps all of MY wines, it is difficult to bring minerality to the bottle without the wine undergoing at least some degree of reduction. I might even say in a few number of cases that the juice or wine did not exhibit any minerality until the wine became at least somewhat reduced. Does this mean that the reduction caused us to sense minerality? Perhaps in some cases. In all cases? No.
I am much more inclined to say that in the majority of my wines, and perhaps in the totality, that reduction performs two roles in relation to our sensing minerality: 1) it underlines or emphasizes a complex of sensations that depend on fruit and the juice, not on the fermentation or maturation; 2) it neutralizes other characteristics (e.g. fruitiness) that would otherwise mask the minerality that is already there.
Lastly, it is important to see that even though supplying plentiful oxygen during the early stages of reduction can combat or erase it, reduction and oxidation are not strictly opposed to each other in winemaking, and certainly one need not cancel out (or "fix") the other. Robert Dentice showed me a quotation from Anselme Selosse where he maintains that the coordinate and conjunct oxidation and reduction of his base wines is essential to his winemaking, and he illustrated the totally common physics of this by pointing to the conjoined oxidation and reduction that take place as a candle burns.

The elaboration of these thoughts has been spurred by Jesse Becker.
Please see his blog.