A subatomic particle is just a particle "smaller than" an atom (hence sub + atomic). The three you're familiar with are the three component particles that make up atoms: protons, neutrons, and electrons. It turns out that protons and neutrons (but not electrons, so far as we know) have further internal sub-structure; protons and neutrons are each made of three smaller particles called quarks. The only other example you "see" in everyday life is the photon, the particle of light. Others exist, but none of them are long-lived or seen in everyday life under normal conditions.

What they "really are" is more a matter of philosophy than it is of physics. Physics is interested in describing how they behave. And in modern physics, we model the behavior of particles through quantum mechanics.

It turns out that the way we tend to think of particles in our minds - tiny little spheres bounding around and exerting forces on one another - just doesn't correspond to the way particles actually behave.

In quantum mechanics, the underlying "reality" of a particle is something called its wavefunction. Rather than a particle "being in one position", a wavefunction takes every possible position and assigns a single number to each of them. These numbers are complex numbers (that is, they look like something like 0.2 - 0.4i), and the Schrodinger equation tells us how they evolve over time.

Exactly how we interpret these wavefunctions as corresponding to any of the things we observe day to day is a topic of some debate. In a sense, the ultimate underlying reality is always the wavefunction, and the Universe as a whole has one wavefunction that has just been evolving according to the Schrodinger equation since the beginning of time. But since we often want to consider particles in isolation so that we can talk about an electron without talking about the whole Universe, we have to think about how we can "cut off" the electron from the cosmos. In this sense we try to talk about "the electron's" wavefunction, but in reality, even the existence or non-existence of the electron is a statement about the wavefunction of the entire Universe, so by talking about "an electron" at all we are going to introduce some weirdness.

We do this by thinking in terms of probabilities. The electron, if measured, will appear to be in one of several locations, and the probability of it being in each of them is equal to the square of the magnitude of the number the wavefunction assigns to that location. For example, if a position has wavefunction value 0.1 - 0.4i as mentioned earlier, the electron will be in that location with probability (0.1)^2 + (0.4)^2 = 0.17. Since the wavefunction has non-zero values at many points, in this sense the electron "can be in more than one place at once", or more properly, could be measured to be in any one of several different places at random.

Exactly how this corresponds to our conventional notions of position, velocity, etc is a topic of considerable debate, but this approach to modeling particles works well and describes our world accurately (and no non-probabilistic model can do so).

A quantum particle isn't exactly a solid chunk of stuff. In some ways, it's more accurately pictured as a ripple in the fields that make up the universe.

It so happens that these ripples only come in whole-number sizes (they are "quantized") — so you can have no electron, or one electron, or two, etc., but you can't have half an electron or seven-fifteenths of an electron.

(That's what makes it quantum physics. In classical physics, there was never any rule that you couldn't split a chunk of matter down indefinitely. That turned out to just be wrong.)

What are "fields", though? A field is just some quantity measured at each point in some space and time. An example of a non-quantum field is the wind on the surface of a planet. The wind has some value at every point on the globe. (It might be zero, if the air is still.) And it changes over time. A breeze and a tornado are two different phenomena in the wind field.

(The wind is a vector field; at any point, it has both a magnitude and a direction. Air pressure and temperature are scalar fields; at any point, each has only a single numerical quantity. Gravity is a vector field; mass density is a scalar field. There are other more complicated kinds of field too.)

Why does it matter that particles are ripples and not tiny solid chunks? Well, one reason is that a ripple is inherently spread out, whereas a solid chunk has an exact single location.

And while solid chunks can only interact by bumping into each other, ripples can interact through interference — they can reinforce or cancel each other out. Solid classical-physics chunks can't do that.

(For another non-quantum example: Interference between ripples is also how, for instance, noise-cancelling headphones work: they make a sound wave that cancels out the noise.)

It's this interference that explains a lot of the "weird" effects in quantum physics, like the double-slit experiment. It's not that a particle "really is" a solid chunk that's magically in two places at once; rather, it never was a solid chunk at all, but a ripple that can interfere with other ripples, including its own echoes.

To begin to approach quantum mechanics, you have to accept that on that very tiny scale, the world does not behave in a way that's intuitive to human experience. Everything you see and interact with as a human is in truth just some macro-scale approximation of an incomprehensibly number of complex relationships formed at unthinkably small scales. So with your human intuition set aside, you're ready to wade into QM.

For purposes of your question, it's probably best to start with the double slit experiment, which hails from the 19th century. I'll trim the fat to keep the ELI5 short, but definitely google videos about this experiment to gain a deeper grasp of what it shows and why that's important. But essentially, the double slit experiment shows that photons (light particles) can behave both as particles and as waves. Photons seem to act particle-like when you squeeze them through a narrow slit, but on the other side, it starts to interfere with itself in a wave-like fashion. This was puzzling, and took us a while to figure out.

In the 1920's, the Heisenberg and Schrodinger you've probably heard about got together and kind of cracked the code behind this strange quantum behavior. Heisenberg pieced together that there is an inherent uncertainty when you try to measure a particle's momentum and position. You can get an arbitrarily highly precise measurement on one of them, at a proportional sacrifice to precision on the other. This is the Uncertainty Principal. At the same time, Schrodinger developed his famous equations, which described quantum systems not in terms of discrete particles, but as single "wave functions." A wave function is sort of a probabilistic expression that describes all of the potential states a quantum system, accounting for this uncertainty. The wave function is a sort of sum of all these potential states, weighted by their probability. In that way, the wave function comes to approximate (very, very precisely) the "superposition" behavior of quantum systems, wherein the system behaves like a mix of all potential states it could take. Schrodinger's equations described how such systems evolve in this probabilistic fashion, and experimentation has justified him time and again.

Now, to return to your question, physics is much less concerned with answering "what things are" at a fundamental level and much more interested in answering "what things do." So we can only really answer the first question in terms of the second. To that end, we see that, at a fundamental level, the universe does not behave classically, but instead its constituent particles behave according to the everywhere-at-once oddity of QM. So we can say that fundamental particles are not simple points of mass or energy--they behave according to a much deeper and more vibrant structure.