What is Super Resolution in the Context of Windowing? I have been trying to understand what super-resolution is, in the context of DSP/DIP. What criteria is being maximized/minimized, and why? 

Most of my online searches yield super-res techniques from the optical physics stand point, however I know anecdotally that they have been used in image processing, and in some doppler-radar processing as well. 

One particular instance of Super-Res I remember seeing was in the design of a windowing function, where the main lobe was the same width as that of a boxcar, while simultaneously having very low and fast degrading side lobes, similar to a hamming window. In this sense, the 'super-res' came from the fact that frequencies very close to one another were able to be resolved due to the low main lobe width, and a very high dynamic range. Is this what super-res really amounts to?

Is there an explanation that, (perhaps), likens it to a form of de-noising, or sparsity pursuit? Or is it a different animal altogether?
The term super-resolution is used very loosely nowadays. But I think the following problem was the original idea, or at least, the one that made the term famous.
Suppose you have a scene and several observations of that scene, i.e. several frames of a video with slight camera motion between them. Super-resolution algorithms combine those several observation into one highly resolved image. This is possible because each observation might contain unique information not present in the others.
If you consider a certain frame as the key frame, this process can be interpreted as a MAP estimation since a priori (or extra) information is present in others frames.
Additionally, we can use natural image statistics as extra information to regularize the solution. Total variation regularization is a broadly used technique.
I suggest the paper Sung Cheol Park - Super Resolution Image Reconstruction: A Technical Overview.