It is important to note that you do not need negative mass for an Alcubierre drive to work, but negative mass density. The difference is subtle, but important for actually being able to build a warp drive.
Essentially what the equations say is that to bend space in the appropriate manner for a warp drive (i.e. to expand space behind your spacecraft and compress it in front), you need to be able to sum up all of the matter in a given portion of space and get something along the lines of "-10 kg / m3."
Space-time curvature is caused not simply by the mass of matter, but also by how compact it is in a given volume, and what we're trying to do with a warp drive is to curve space in a very specific way, so it doesn't matter what we're doing with the matter-energy in a given volume, so long as we can make its mass density appear to be negative.
The easiest way to do this is to just take a lump of matter that weighs -10 kg, and put it in your space. This however, doesn't make sense, since how can something "weigh" negative 10 kilos? That's like saying I have negative 3 apples. This is why it is sometimes said we need "exotic matter" to make warp drives work, however, there are a few examples of ways to cheat this.
I'll start with an analog in silicon doping. In order to create integrated circuits, bulk silicon is doped to either be positive or negative charged. The way you do this is by adding impurities to pure silicon to change the number of valence electrons in its structure. To make positively charged silicon, boron is added. Because boron has one fewer valence electron vs. silicon, this results in a deficiency in electrons. Similarly, to make negatively charged silicon, phosphorus is added. Because phosphorus has an extra valence electron vs. silicon, there is a surplus of electrons, leading to a negative charge. Because these electrons are surplus, they're free to move around the silicon lattice, and controlling this flow is how circuits work.
The more interesting thing here, however, is what happens when you consider not the electrons as a particle, but the absence of electrons. In circuit design, we call this a hole, and we treat it as a positively charged particle, one that doesn't actually exist.
In this way, you can actually view this as a negative electron density. Essentially, we have created a positively charged particle using only negatively charged matter! This is very similar to what we want for our warp drive if you think of mass not as an electrical charge, but instead a sort of gravitational charge.
There are many equations that treat mass like a gravitational charge, and it's telling that gravity is unique among the fundamental forces in that it does not, to our knowledge, have opposite charges. This means gravity forms monopoles, and everything we've seen about monopoles suggests that nature abhors them. Really it makes no sense, when viewing mass as a kind of charge, for there to not be the concept of an opposite charge.
This probably indicates our models are incomplete with respect to how we view mass and gravity, and indeed, we are unable to combine gravity with quantum mechanics.
So can we find a little more concrete example of negative mass density? Yes, actually, we can find a few.
The first, as others have mentioned, is the Casimir effect. In a nutshell, what happens in the Casimir effect is that two plates close together are pulled together with an extra, anomalous force (i.e. not due to charge or gravity).
The reason for this comes when we look at empty space from a quantum point of view. Quantum physics says that vacuum isn't really empty, but is a seething mass of "virtual particles" constantly popping into existence in pairs then combining with each other and annihilating. Because they recombine, the energy density of the vacuum balances back out to zero (in fact, that it exactly cancels out to zero is considered one of the great unsolved mysteries of physics).
So what does this have to do with the Casimir effect? Well, if space is empty, the vacuum energy can easily cancel back out to zero as virtual particles pop into existence, do their thing, and pop out of existence. However, things change once we add mass (i.e. our two plates).
Virtual particles are treated as a spectrum. As far as the math is concerned, we treat it as an infinite sum of particles that all exactly cancel out to zero. Because the virtual particle spectrum has infinite energies, per De Broglie it also has infinite wavelengths. This is where things start to get interesting once we add our plates.
For a particle to fit between those plates, it must have a wavelength smaller than the distance between them. Now, because our plates are close together, they block some of these virtual particles from popping into existence, namely those with a larger wavelength than the gap between them. Now, more particles of the larger wavelength are forming outside the plates than in between them. What this ultimately means is we've altered the density of virtual particles in space. Because we can say these virtual particles have mass (even if it is short-lived), we have now changed the mass density of the space in between and around the plates. So if normal, empty space has a mass density of zero, what does that mean for the mass density between the plates? Because it has less of a density than the space outside the plates, and that space has a density of essentially zero, it must, for all intents and purposes have negative mass density. We've just created a form of negative mass! The Casimir force however, is incredibly weak, as a particle's energy is directly related to its wavelength. In order to increase the Casimir force, you must decrease the separation of the plates (and thus block shorter wavelength, higher energy virtual particles), and so you can only create very small amounts of negative mass density with it.
Just like our silicon doping example, we've created a negatively charged attribute just from manipulating the distribution of a positively charged one. We have, in effect, "doped the vacuum." Unfortunately, we can't really use this for creating our warp drive, but it does at least show us that negative mass density is possible.
As a sidenote, this "pairing" of positive and negative attributes cancelling out to zero was first envisioned in a vacuum model known as the Dirac Sea. This model basically states that the vacuum is an infinite sea of negative energy particles (hey, that's exactly what we want), and although it turned out to be not true, it did make valuable scientific predictions that were later confirmed, namely the existence of antimatter. Even though the Dirac Sea view is incomplete with regards to the vacuum, it is directly analogous to our silicon doping example.
So are there more useful examples of negative mass density in nature we can use? Yes, it turns out there are. Two of them, in fact.
I will start with the more scientifically accepted version first, one that we can see all around us: The accelerating expansion of the Universe.
It is generally agreed, based on astronomical observations, that the Universe's expansion is accelerating. This creates a bit of a problem: something must power this acceleration, and that means energy. So to make our equations match our observation, we add some energy to the vacuum, making it expand faster.
We treat this as an intrinsic property of space, so as space expands, its density doesn't decrease. That means adding more space is also adding more energy! This creates a new problem: energy means mass, and mass means gravity. If we're adding more and more energy to the Universe as it expands, we're adding more and more mass! The Universe shouldn't be expanding at all then, the energy making it expand faster and faster should itself generate gravity that slows the expansion!! We call this strange energy that seems to come from nowhere "dark energy" because we don't understand what it is at all.
So what do we do about this problem? Mass and energy curve space, so adding more energy should curve space more, and indeed it does (the Universe accelerates faster), but the space is curving in the wrong direction since the Universe is speeding up, not slowing down! Clearly whatever "dark energy" is, it's not "normal" energy in any way.
Looking at things from a general relativity standpoint, the accelerating expansion means that empty space has negative curvature. This may not sound weird, but what it's basically saying is that the negatively-curving space has a negative mass-energy density! All of our data is telling us at every point in the entire Universe, we have negative-mass! We have exactly what we need to build our warp-drive, everywhere, but we don't have the slightest clue what it is! It's like the Universe itself is taunting us!
But once again, it shows us that negative mass is, in fact, a real, natural thing. The only question is how do we create this negative mass in the configuration we need to go really fast?
Our last, best hope lies in a very controversial conjecture of physics: The Woodward effect.
Essentially, the Woodward effect says that if you accelerate any object that is absorbing or discharging energy, that you can create a transient change in the mass of that object.
In fact, it is theorized that this transient change can even be negative!
So what kinds of systems can absorb energy and release it that we can also easily accelerate? Well, one of the simplest ways to model such a system is using capacitors and inductors. These are objects that are simple, cheap, and can absorb and discharge energy rapidly without relying on mechanical components.
This leads us to our modern day research into warp fields. Nearly every warp field experiment is using capacitors in some kind of configuration designed to make use of the Woodward effect. Often these capacitor banks take the form of a ring, this is because you can maximize the bending of space inside the ring, thus making it easier to detect.
It is perhaps also telling that fringe theories such as the EmDrive rely on microwave resonant cavities, which themselves behave as a capacitor and inductor tied together. In other words, an EmDrive can be modeled as the exact system predicted to create negative mass by the Woodward effect!
In the end, if it turns out the Woodward effect is real (big if), then we have a way of generating negative mass density, and interestingly enough, you can probably do it with a microwave resonant cavity (which is essentially what an EmDrive is). I find it interesting how all of these seemingly unrelated fringe ideas seem to be converging at the same point.
So why do NASA's artistic depictions of Alcubierre drive spaceships use rings?
This is a very different question, but one still related to negative mass density. When the Alcubierre drive was first proposed, many scientists did some calculations and decided that even if it were possible to construct negative mass, that you'd need planet-sized amounts of this negative mass to create a warp field (on the order of the mass of Jupiter).
This is clearly unrealistic, and seemed to mean bad news for our plans to go fast, but some other scientists did more calculations and determined that no, you don't need planet-sized amounts of negative mass, you can just do it with less than a metric ton.
So why the huge discrepancy? It all boils down to the topology of your warp field. The scientists who came up with the planetary mass estimate created a warp field that was essentially spherical in shape, whereas the scientists who came up with the much smaller estimate devised a warp field that is toroidal shaped.
But it gets more interesting than that: the speed at which you can travel faster than light, known in warp research fields as the boost velocity, is directly proportional to how thick your warp bubble is. It becomes very difficult to shrink the thickness of your warp bubble since it requires increasing the mass density along the edge of your warp bubble. But in doing so, you minimize the amount of exotic mass you need while maximizing the amount of boost you can get. It turns out the easiest way to create a thin, yet dense edge to your warp bubble like this is to make your bubble the shape of a torus.
Since the warp field is a function of how you arrange your negative mass density, that's where your rings come from.
So do the rings contain the exotic matter? Well, yes and no. As we've already established, you don't need actual matter weighing negative 10 kilos, you need negative mass density, and as we've shown before, it should be possible to create such a mass density using normal matter.
It is therefore more appropriate to say that the rings contain the mechanism for creating negative mass density, and most likely that negative mass density exists as a field around the rings.
To finish my answer, I will leave you with a world-building scenario that allows you to create a warp-drive with minimum hand-waving:
Let's say your warp drive works by taking energy and using it to boost the appropriate type of virtual particle into existence. The how it does this is the only hand-waving part we have.
Virtual particles are everywhere in empty space. They are represented as an infinite spectrum of properties, and always appear in pairs and annihilate, returning the average energy density of the vaccuum to zero. So long as the particles "pay back" the energy for their existence by recombining, they can literally have any characteristics, as long as the pair allows them to combine back to zero. In general, we might say the pair have opposite charges, but the equations work just as well to say the particles have "opposite masses," so long as everything sums up to zero energy in the end.
What our warp drive does then is use energy to force virtual particles that have negative mass into existence. It does this by "paying" for the energy that would have otherwise been paid by the virtual particles annihilating. This is analogous to Hawking radiation, where a black hole "pays" the Universe for a particle in a virtual particle pair, causing one of the virtual particles to be boosted into reality. This particle then escapes the black hole, carrying energy away from it.
So by specifically tuning how we move energy around our spaceship (hand-wave part), like a black hole, we too boost virtual particles into existence, but only those with negative-mass, and use them to form our warp bubble. This warp bubble causes space in front of our ship to compress, and space behind it to expand in a manner similar to how the Universe expands. Because there is no limit to how fast space can expand, there is essentially no limit to how fast we can go. Inside our warp bubble, everything is stationary, but to an outside observer, we seem to be moving faster than light. Because we aren't actually moving, time dilation is only a problem at the infinitesimally small edge of our warp bubble.
This process takes constant energy input, as these negative mass particles want to decay to more stable forms (and in the process presumably acquire positive mass), but they exist on timescales long enough for us to create a warp bubble. Without that energy input, our warp field collapses, and we're stuck at sublight speeds.
