The main purpose of this talk is to explain and explore the object in the title and to outline why it might be useful. In particular, I hope to organize the following questions: when can (chromatic-type) homology theories be realized by structured ring spectra and, if they can, what can you say about maps between them? Both problems can be formulated in terms of an Andre'-Quillen cohomology calculation, which is where the tangent complex comes in. With any luck, I will get to the point where I can talk about some of the applications to elliptic spectra arising from the moduli stack of elliptic curves. I am, of course, following closely in the footsteps of others, in particular of Mike Hopkins and Haynes Miller.