Question: My understanding is limited and I'm trying to learn more about how homotopy forms the notion of equivalence. I can grasp its definition as "continuous", but my understanding of homotopy falls away in spaces that are discrete. How does the mapping actually work?

Answer: 
Two discrete spaces are homotopy equivalent if and only if they have the same number of points. Any function whose domain is discrete is automatically continuous, and two functions $X \to Y$ for discrete $Y$ are homotopic if and only if they are equal.

(I'm not sure I understood your question, let me know if that clears things up or not.)
Old question, but let me cast some light (or maybe not) for future readers.
Instead of looking at homotopy in discrete spaces, it works better looking to define a homotopy between graphs, so you could identify the form, or at least the connections between discrete objects, graphs inherit a topology by it's connection to groups, so it shouldn't be that hard. 
If I could say that something involving graphs is easy, which I can not, but like I've said before, this is just to cast some light in the subject.