At least when it comes to physical reality, which I define here as that which exists in the cosmos, there is no such thing as complete emptiness.
Quite the opposite, it seems that the more we learn about nature, the busier space becomes. We can, of course, contemplate the idea of a metaphysical emptiness, a complete void where there is nothing, what some people like to call absolute nothingness. But these are concepts we make up, not necessarily things that exist. In fact, calling nothingness a "thing" automatically makes it into a something, a curious paradox.
This is why Leucippus and Democritus, the pair of pre-Socratic Greek philosophers credited with the invention of the atomic concept — that everything is made of tiny bits of stuff — suggested the joint existence of atoms and the void: Atoms make up everything that exists, but they move in complete emptiness — the void.
It's useful, as an exercise in the ever-evolving way we figure things out about the world, to make a list of the things that fill up empty space — at least as of today. (The list does change. For example, 120 years ago it would include the aether, the medium where light was supposed to propagate.) Starting with classical physics, the key concept is that of a field. A field is an amazing idea, a sort of spatial manifestation of a source of some kind. If an object sensitive to the field is placed within its range, it will respond in some way — usually, either by being attracted to or repelled by the source that creates the field.
In classical physics we know of only two forces, gravitational and electromagnetic. Every object with mass attracts every other object. You attract and are attracted by everyone that exists (even your worst enemies), by the butterflies and the whales, by the sun and all the planets of this solar system and all others across the entire universe. The intensity of the gravitational field of an object grows in proportion to its mass and decays with the square of the distance to it.
That gravity is a very weak force is clear: Just think that the magnetic attraction of a small fridge magnet beats the whole Earth pulling on it gravitationally. Still, gravitational fields extend their threads to all corners of space. Since fields carry energy, we can say that space itself is filled with the energy of these gravitational fields. (Electromagnetic fields also have energy, of course. But since electric and magnetic forces can be both attractive and repulsive, they usually are neutralized and rarely manifest themselves at large distances. Exceptions are, for example, magnetic storms in stars and the magnetic field of the galaxy.)
At the quantum level, things get even more dramatic. Since in the world of atoms and subatomic particles nothing stands still, there is an energy associated with this residual motion called "zero-point energy" or "vacuum energy." There is no such thing as zero energy. If we now connect this fact to the famous E=mc2 formula, which states that energy and matter may be inter-convertible (with subtleties that I will overlook here), it is possible for particles of matter to spring out from the energy of the vacuum, the energy of "empty space." That matter may come out of nothing serves to show that the nothing of quantum physics is far from a complete void. These "virtual" particles appear and disappear like bubbles in a boiling soup. In the current view of quantum physics, the void bubbles incessantly with the creation and destruction of matter particles.
The concept of fields carries over to quantum physics with even more dramatic effects. We actually don't refer to particles anymore, but to the fields that originate them: An electron or a proton are excitations of the electron or the proton fields, respectively, like small waves on the surface of a lake. Particles are pictured as moving knots of energy in their fields, with physical properties like mass and energy. The physical picture that emerges is that of space filled with quantum fields that boil up with real and virtual particles.
The recently discovered Higgs is a famous example of a quantum field. Even if the Higgs particle has a mass about 126 times larger than that of a proton, it's still seen as a lump of the Higgs field. The field associated with this particle permeates all of space, somewhat like the air in our atmosphere. Other fields, like that of the electron, feel the Higgs field and interact with it. It's from these field-field interactions that the electron particle (and all others particles we know) gains its mass.
As the Fox said to the Little Prince, "What is essential is invisible to the eyes." This is a very accurate description of the worldview according to modern physics.
Marcelo Gleiser's latest book is The Island of Knowledge: The Limits of Science and the Search for Meaning. You can keep up with Marcelo on Facebook and Twitter: @mgleiser.