My favorite elegant explanations will already have been picked by others who turned in their homework early. Although I am a theoretical physicist, my choice could easily be Darwin. Closer to my area of expertise, there is General Relativity: Einstein's realization that free-fall is a property of space-time itself, which readily resolved a great mystery (why gravity acts in the same way on all bodies). So, in the interest of diversity, I will add a modifier and discuss my favorite annoying elegant explanation: quantum theory.
As explanations go, few are broader in applicability than the revolutionary framework of Quantum Mechanics, which was assembled in the first quarter of the 20th century. Why are atoms stable? Why do hot things glow? Why can I move my hand through air but not through a wall? What powers the sun? The strange workings of Quantum Mechanics are at the core of our remarkably precise and quantitative understanding of these and many other phenomena.
And strange they certainly are. An electron takes all paths between the two points at which it is observed, and it is meaningless to ask which path it actually took. We must accept that its momentum and position cannot both be known with arbitrary precision. For a while, we were even expected to believe that there are two different laws for time evolution: Schrödinger's equation governs unobserved systems, but the mysterious "collapse of the wave function" kicks in when a measurement is performed. The latter, with its unsettling implication that conscious observers might play a role in fundamental theory, has been supplanted, belatedly, by the notion of decoherence. The air and light in a room, which in classical theory would have little effect on a measuring apparatus, fundamentally alter the quantum-mechanical description of any object that is not carefully insulated from its environment. This, too, is strange. But do the calculation, and you will find that we used to call "wave function collapse" need not be postulated as a separate phenomenon. Rather, it emerges from
Schrödinger's equation, once we take the role of the environment into account.
Just because Quantum Mechanics is strange doesn't mean that it is wrong. The arbiter is Nature, and experiments have confirmed many of the most bizarre properties of this theory. Nor does Quantum Mechanics lack elegance: it is a rather simple framework with enormous explanatory power. What annoys me is this: we do not know for sure that Quantum Mechanics is wrong.
Many great theories in physics carry within them a seed of their demise. This seed is a beautiful thing. It hints at profound discoveries and conceptual revolutions still to come. One day, the beautiful explanation that has just transformed our view of the Universe will be supplanted by another, even deeper insight. Quantitatively, the new theory must reproduce all the experimental successes of the old one. But qualitatively, it is likely to rest on novel concepts, allowing for hitherto unimaginable questions to be asked and knowledge to be gained.
Newton, for instance, was troubled by the fact that his theory of gravitation allowed for instant communication across arbitrarily large distances. Einstein's theory of General Relativity fixed this problem, and as a byproduct, gave us dynamical spacetime, black holes, and an expanding universe that probably had a beginning.
General Relativity, in turn, is only a classical theory. It rests on a demonstrably false premise: that position and momentum can be known simultaneously. This may a good approximation for apples, planets, and galaxies: large objects, for which gravitational interactions tend to be much more important than for the tiny particles of the quantum world. But as a matter of principle, the theory is wrong. The seed is there. General Relativity cannot be the final word; it can only be an approximation to a more general Quantum Theory of Gravity.
But what about Quantum Mechanics itself? Where is its seed of destruction? Amazingly, it is not obvious that there is one. The very name of the great quest of theoretical physics—"quantizing General Relativity"—betrays an expectation that quantum theory will remain untouched by the unification we seek. String theory—in my view, by far the most successful, if incomplete, result of this quest—is strictly quantum mechanical, with no modifications whatsoever to the framework that was completed by Heisenberg, Schrödinger, and Dirac. In fact, the mathematical rigidity of Quantum Mechanics makes it difficult to conceive of any modifications, whether or not they are called for by observation.
Yet, there are subtle hints that Quantum Mechanics, too, will suffer the fate of its predecessors. The most intriguing, in my mind, is the role of time. In Quantum Mechanics, time is an essential evolution parameter. But in General Relativity, time is just one aspect of spacetime, a concept that we know breaks down at singularities deep inside black holes. Where time no longer makes sense, it is hard to see how Quantum Mechanics could still reign. As Quantum Mechanics surely spells trouble for General Relativity, the existence of singularities suggests that General Relativity may also spell trouble for Quantum Mechanics. It will be fascinating to watch this battle play out.