Ah, I know this one!
# What's in a proton?
A proton is really made of _quantum fields_. Remember that. Any time you hear any other description of the composition of a proton, it's just some approximation of the behavior of quantum fields in terms of something people are likely to be more familiar with. We need to do this because quantum fields behave in very nonintuitive ways, so if you're not working with the full mathematical machinery of [QCD](http://en.wikipedia.org/wiki/Quantum_chromodynamics) (which is hard), you have to make some kind of simplified model to use as an analogy.
One of the more confusing things about quantum fields is that they react differently depending on how they are observed. The way we observe protons is by hitting them with other high-energy particles in a particle accelerator and seeing what comes out. In a slow collision, with very little energy involved, the proton acts like a single point particle. If we give the particles slightly more energy, the proton looks more like a blob with three points in it --- these are the three quarks of common knowledge. (Incidentally, the reason you see images like the one you found on Wikipedia is that for a long time, people were colliding protons at the intermediate energies where they appear to behave as a group of three quarks.) If we give the colliding particles even more and more energy, the proton will appear to be an ever-more-dense amalgamation of all sorts of particles: quarks, antiquarks, gluons, photons, electrons, and everything else. We call these particle _partons_ (because they're part of the proton).
This diagram shows basically how a proton appears to different kinds of collisions. The vertical axis basically corresponds to collision energy, and the horizontal axis corresponds to the "resolving power" of the incident ("probe") particle. (Resolving power is basically transverse momentum, but I can't explain how that connection works without getting into more detail of quantum mechanics than I think is necessary.) The contents of each circle represents, roughly, a "snapshot" of how the proton behaves in a collision at the corresponding energy and resolving power.
[![kinematic diagram of proton composition][1]][2]
So for example, if you hit a proton with a beam of high-energy probes that have weak resolving power, it behaves like a dense cluster of partons (quarks and gluons etc.), each of which is fairly large. That corresponds to the top left corner of the graph. Or if you hit the proton with a beam of low-energy probes with high resolving power, it behaves like a sparse cluster of partons, each of which is small. If you hit it with a beam of low-energy, low-resolving-power probes (but not _too_ low), it behaves like a collection of three particles, as seen in the bottom left corner.
Physicists describe this apparently-changing composition using parton distribution functions, $f(x, Q^2)$, which under certain conditions can be interpreted as the probability density of the probe interacting with a particular type of parton with a particular amount of momentum. Basically, $f(x, Q^2)$ is related to the number of particles in the circle at the corresponding $(x,Q)$ point on the plot. For more information on parton distributions, I would refer you to [this answer of mine](http://physics.stackexchange.com/a/13866/124) and the resources named therein, as well as [this one](http://physics.stackexchange.com/a/22212/124).
# What's the gray region?
In the preceding image, I displayed each snapshot of the proton as a set of partons (quarks and gluons etc.) uniformly distributed within a circle, as if the proton has a definite edge. But in reality, that's not the case; instead, the quantum fields that make up a proton gradually fade away to zero as you move further away from the center. It has a fuzzy edge. So a (somewhat) more accurate snapshot would look something like this:
[![proton without a sharp edge][3]][4]
Notice that there are more partons near the center of the proton, and progressively fewer as you move toward the edge. The ordinary parton distributions that I mentioned above, $f(x, Q^2)$, are part of a simplified model in which we ignore this fact and pretend that partons are distributed uniformly throughout space. But we can make a more complicated model that _does_ take into account the fact that partons are clumped up toward the center of the proton. In such a model, instead of regular parton distributions, you get more complicated functions, called impact parameter-dependent parton distributions, and denoted $f(x, Q^2, b)$, where $b$ is the radial distance - the impact parameter.
There have been some theoretical studies showing that these impact parameter-dependent parton distributions trail off gradually as you go to large radii. For example, see figure 5 of [this paper](http://prd.aps.org/abstract/PRD/v83/i3/e034015) ([arXiv](http://arxiv.org/abs/arXiv:1010.0671)) or figure 7 in [this one](http://prd.aps.org/abstract/PRD/v84/i9/e094022) ([arXiv](http://arxiv.org/abs/arXiv:1106.5740)):
Here $N(y)$ is a quantity related to the parton distributions (specifically, it's the color dipole scattering amplitude), which kind of "condenses" the many different parton distributions into one quantity. (Huge oversimplification, but it's good enough for this.) You can then define the spatial extent of the proton as the region in which $N(y)$ is above, say, 5% of its maximum value. Or 10%. Or 50%. The exact number is somewhat arbitrary, but the point is, whatever number you pick, you'll wind up with a circle that encompasses the region in which the parton density is high, kind of like this:
[![proton with fuzzy edge and with a circle behind it][5]][6]
This is roughly what the gray circle in the image from Wikipedia represents. It's a region with a size on the order of $1\text{ fm} \approx 5\text{ GeV}^{-1}$, where the chance of an incident particle (a probe) scattering off the proton is relatively significant. Equivalently, it's the region in which the parton distributions are large, and also the region in which the quantum fields that constitute the proton are much different from zero.
As you can guess, all this is pretty imprecise. You _can_ make a more rigorous definition of the size of a proton by using the [scattering cross section](http://en.wikipedia.org/wiki/Scattering_cross-section). You can also get a definition without using scattering, using the [charge radius](http://en.wikipedia.org/wiki/Proton#Charge_radius), which can be measured or calculated using various other methods. I won't go into those, as the details would be material for a whole separate question, but the results of all these methods come out to a radius a little less than $1\text{ fm}$.
---
Incidentally, this claim of a proton being 99% empty space is probably false using any reasonable definition. You might be thinking of _atoms_, where the volume in which the electron's quantum field has an appreciable value is much larger than the size of the electron itself, whatever it may be. People sometimes simplify that to say that the atom consists of a large fraction of empty space. But you can't really do the same with a proton, given the large number of particles in it and the strength of their interactions.
[1]: https://i.sstatic.net/h87EN.png
[2]: https://i.sstatic.net/h87EN.png
[3]: https://i.sstatic.net/8pGMK.png
[4]: https://i.sstatic.net/8pGMK.png
[5]: https://i.sstatic.net/7eRzO.png
[6]: https://i.sstatic.net/7eRzO.png