I have had problems with mold in my kitchen behind the sink cabinet a couple of times. I have gutted the sink cabinet, the drywall behind it and the lazy susan area, leaving the countertop in place. The countertop is new and I don't want to break it. I want to replace the drywall with a mold retardant surface. That is why I want to know what greenboard is before I rebuild the cabinet under the sink.
Greenboard is sheetrock where the paper is water resistant (but not necessarily water proof). A better alternative to greenboard is to use a paperless sheetrock product like Densarmor, which uses a fiberglass backing rather than paper (it's the paper that mold likes).
Note that these products aren't designed for wet applications, however. At best, they are designed for areas that 'are sometimes damp'. If you have mold issues under your sink, the problem is water. You have a leak somewhere. That has to be fixed. Replacing the sheetrock--even with one of these other products is just a bandaid and the problem will return if the leak isn't fixed.
A lot of people have issues behind cabinets because the drywall isn't painted properly. And if things are painted properly then it usually doesn't matter. Hit it with two coats of primer and one strong oil based paint.
If you need truly mold proof you need to either you go hardieboard or you can use a waterproofing paint - these are expensive and a pain to apply.
I refuse to use Green board if there is even a chance of wetness and when the wall will be rarely seen (ie under steps in garage, in cabinets). Hardiebacker board or cement board is the ideal product. 1/2 inch cement board is available to go flush with drywall. It is Mold proof and strong as heck. You can always paint it after too.