The Higgs Mechanism
The beauty of science—in the long run—is its lack of subjectivity. So answering the question "what is your favorite deep, beautiful, or elegant explanation" can be a little disturbing to a scientist, since the only objective words in the question are "what", "is", "or", and (in an ideal scientific world) "explanation." Beauty and elegance do play a part in science but are not the arbiters of truth. But I will admit that simplicity, which is often confused with elegance, can be a useful guide to maximizing explanatory power.
As for the question, I'll stick to an explanation that I think is extremely nice, relatively simple (though subtle), and which might even be verified within the year. That is the Higgs mechanism, named after the physicist Peter Higgs who developed it. The Higgs mechanism is probably responsible for the masses of elementary particles like the electron. If the electron had zero mass (like the photon), it wouldn't be bound into atoms. If that were the case, none of the structure of our Universe (or of life) would be present. That's not the correct description of our world.
In any case, experiments have measured the masses of elementary particles and they don't vanish. We know they exist. The problem is that these masses violate the underlying symmetry structure we know to be present in the physical description of these particles. More concretely, if elementary particles had mass from the get-go, the theory would make ridiculous predictions about very energetic particles. For example, it would predict interaction probabilities greater than one.
So there is a significant puzzle. How can particles have masses that have physical consequences and can be measured at low energies but act as if they don't have masses at high energies, when predictions would become nonsensical? That is what the Higgs mechanism tells us. We don't yet know for certain that it is indeed responsible for the origin of elementary particle masses but no one has found an alternative satisfactory explanation.
One way to understand the Higgs mechanism is in terms of what is known as "spontaneous symmetry breaking," which I'd say is itself a beautiful idea. A spontaneously broken symmetry is broken by the actual state of nature but not by the physical laws. For example, if you sit at a dinner table and use your glass on the right, so will everyone else. The dinner table is symmetrical—you have a glass on your right and also to your left. Yet everyone chooses the glass on the right and thereby spontaneously breaks the left-right symmetry that would otherwise be present.
Nature does something similar. The physical laws describing an object called a Higgs field respects the symmetry of nature. Yet the actual state of the Higgs field breaks the symmetry. At low energy, it takes a particular value. This nonvanishing Higgs field is somewhat akin to a charge is spread throughout the vacuum (the state of the universe with no actual particles). Particles acquire their masses by interacting with these "charges." Because this value appears only at low energies, particles effectively have masses only at these energies and the apparent bottleneck to elementary particle masses is apparently resolved.
Keep in mind that the Standard Model has worked extremely well, even without yet knowing for sure if the Higgs mechanism is correct. We don't need to know about the Higgs mechanism to know particles have masses and to make many successful predictions with the so-called Standard Model of particle physics. But the Higgs mechanism is essential to explaining how those masses can arise in a sensible theory. So it is rather significant.
The Standard Model's success nonetheless illustrates another beautiful idea essential to all of physics, which is the concept of an "effective theory." The idea is simply that you can focus on measurable quantities when making predictions and leave understanding the source of those quantities to later research when you have better precision.
Fortunately that time has now come for the Higgs mechanism, or at least the simplest implementation which involves a particle called the Higgs boson. The Large Hadron Collider at CERN near Geneva should have a definitive result on whether this particle exists within this coming year. The Higgs boson is one possible (and many think the most likely) consequence of the Higgs mechanism. Evidence last December pointed to a possible discovery, though more data is needed to know for sure. If confirmed, it will demonstrate that the Higgs mechanism is correct and furthermore tell us what is the underlying structure responsible for spontaneous symmetry breaking and spreading "charge" throughout the vacuum. The Higgs boson would furthermore be a new type of particle (a fundamental boson for those versed in physics terminology) and would be in some sense a new type of force. Admittedly, this is all pretty subtle and esoteric. Yet I (and much of the theoretical physics community) find it beautiful, deep, and elegant.
Symmetry is great. But so is symmetry breaking. Over the years many aspects of particle physics were first considered ugly and then considered elegant. Subjectivity in science goes beyond communities to individual scientists. And even those scientists change their minds over time. That's why experiments are critical. As difficult as they are, results are much easier to pin down than the nature of beauty. A discovery of the Higgs boson will tell us how that is done when particles acquire their masses.