Furry Rubber Bands
I learned electrodynamics from Mark Heald and his concise text on an even more concise set of equations, Maxwell's. In 4 lines, just 31 characters (or less with some notational tricks), Maxwell's equations unified what had appeared to be unrelated phenomena (the dynamics of electric and magnetic fields), predicted new experimental observations, and contained both theoretical advances to come (including the wave solution for light, and special relativity) and technologies to come (including the fiber optics, coaxial cables, and wireless signals that carry the Internet).
But the explanation that I found to be so memorable was not Maxwell's of electromagnetism, which is well known for its beauty and consequence, it was Mark's that electric field lines behave like furry rubber bands: they want to be as short as possible (the rubber) but don't want to be near each other (the fur). This is an easily-understood qualitative description that has served me in good stead in device design. And it provides a deeper, quantitative insight into the nature of Maxwell's equations: the local solution for the field geometry can be understood as solving a global optimization.
These sorts of scientific similarities that are predictive as well as descriptive help us reason about regimes that our minds didn't evolve to operate in. Unifying forces is not an everyday occurrence, but explaining them can be. Recognizing that something is precisely like something is a kind of object-oriented thinking that helps build bigger thoughts out of smaller ideas.
I understood Berry's phase for spinors by trying to rotate my hand while holding up a glass; I mastered NMR spin echoes by swinging my arms while I revolved; the alignment of semiconductor Fermi levels at a junction made sense when explained as filling buckets with water. Like furry rubber bands and electric fields, these relationships represent analogies between governing equations. Unlike words, they can be exact, providing explanations that connect unfamiliar formalism with familiar experience.