The Greek atomists were the first to ponder the fundamental constitution of matter. They considered, in an amazingly prescient insight, that if you could cut matter into smaller and smaller pieces you'd end up with its smallest bits, which they called atoms.
The word itself means that which cannot be cut. They further considered that atoms were eternal and indestructible, thus constituting the essence of Being. However, as they combined with each other in myriad ways, they made up all the stuff that we see in the world, from rocks to water drops to frogs and people. This way, Being turns into Becoming, capturing the essence of nature, of things that are and things that change. They went further, and in an attempt to create a unified theory of nature, proposed that thoughts and feelings were also made of atoms. Unified theories are as old as philosophy.
Although the modern concept of atoms is quite different from that of the pre-Socratic Greeks, the notion that matter is made up of small, indivisible bits remains alive and well, constituting the basis of elementary particle physics, the branch of physics that tries to find the fundamental constituents of matter.
It all started with J.J. Thomson who, in 1897, found the electron. People knew that atoms existed (or at least they conjectured they did, as chemists had made quite clear), but they were surprised to find out that they were divisible. Thomson showed that the electron was about 2,000 times lighter than the hydrogen ion, which was the lightest thing around. (Hydrogen, as you recall, is the simplest atom in nature, with a proton in its nucleus and an electron in orbit around it. The hydrogen ion is simply the atom stripped of its electron, which, for hydrogen, means a lone proton.)
In 1911, Ernest Rutherford showed that the atomic nucleus was small and dense. Then, in 1913, Niels Bohr came up with his atomic model, where electrons move about the nucleus in circular orbits separated from each other like rungs in a ladder: electrons were allowed to jump from one orbit to another, but not to be in between them.
The twentieth century saw the number of small bits of matter multiply a hundred fold. As experiments became more sophisticated and probed matter at higher and higher energies, all sorts of "elementary" particles started to show up. A remarkable trade off between energy and matter was at play, the expression of Einstein's famous E=mc2 formula: if you accelerate bits of matter to very high energies, making them collide head on, the energy of their motion can transform into new bits of matter. The rules that control these matter transmutations are the most basic laws of nature, in the sense that the total amount of the conserved quantity remains the same before and after the collision: conservation of energy, conservation of electric charge, and a bunch of other conserved quantities. In a sense, these laws constitute the Being of modern physics, while the myriad material transmutations constitute the Becoming.
After much searching, by the 1980s a model capturing all that was known in particle physics emerged, the Standard Model. Remarkably, all the hundreds of particles discovered during the previous decades were made of only twelve: six quarks (that make up protons, neutrons and a bunch of other "hadronic" particles) and six leptons (the electron, the muon, the tau and their respective neutrinos). This amazing synthesis is the crown jewel of modern high-energy physics: everything that we see in nature is composed of these 12 particles.
These are the current smallest bits of things. Are they THE smallest bits of things? That is, could there be another layer of matter underneath the one we are currently aware of, where these 12 particles (and the Higgs, of course, the elusive particle being tracked down at CERN in Switzerland) are seen as combinations of other, more fundamental stuff?
All we can say right now is that our current theories and experiments don't see any indication that this is the case. However, there are many questions that the current Standard Model doesn't address, and it's quite possible — almost certain — that new particles and theories will emerge from the cracks of our current understanding.
For example, neutrinos have been shown to have mass, something the Standard Model doesn't address. Also, most matter in the universe is in the form of dark matter, which seems to exist in a 6:1 proportion in relation to ordinary atoms. We don't know what dark matter is, apart from knowing it's not made of protons and electrons. I also haven't said anything about the particles that transmit the forces between the particles of matter, the photons of electromagnetic interactions, and the gluons and weak bosons of the strong and weak nuclear force. On another day we'll go into those and the concept of fields.
We have come a long way from the Greeks. But what we do today carries the essence of their thinking. Nature is complex, and one way of making sense of it is by dividing it into small bits. It may be best to call them the smallest current bits of matter, instead of THE smallest bits.
Scientific questions that ask for final answers are, by definition, unanswerable.