Current physics consider matter to be fundamental, so breaking it down should eventually reveal fundamental particles, and it did. Electrons and quarks are examples and even a photon, which has no mass or charge, is said to be a fundamental particle. This method of breaking apart matter to find out what it is made of requires billion-dollar particle accelerators, but it began with the atom.
Initially, atoms were thought to be indivisible, like little solid billiard balls, until Lord Rutherford showed they weren’t by firing alpha particles at a piece of gold foil, when they mostly went straight through and only a few bounced back. It turned out that 99.9999…% of the mass of the atom is in its nucleus, and the rest is just a cloud of tiny electrons whizzing about.
Bohr then suggested that the atom is like a solar system, but held together by electrical forces not gravity. This worked for a while but electrons, which are matter, routinely pass right through each other in a way that planets don’t. Two different electrons can also occupy the same orbit, which again planets can’t, and while planet orbits are elliptical, electron orbits are perfectly spherical. The atom isn’t like a tiny solar system! The modern view is based on what is called the standard model of physics, which summarizes everything physicists have learned about matter.