Practically all matter is composed of atoms, which are extremely small particles. Atomos is greek and means "indivisible", because the Greek philosopher Democritus, who first thought of the concept, thought that they were. But they are not indivisible. They have a complex internal structure, which was discovered in 1911 by the physicist Ernest Rutherford.
Rutherford did an experiment where he fired alpha particles (helium nuclei) at a very thin gold foil. At that point, it was thought that atoms were solid spheres of matter, as detailed in the theories of John Dalton. Therefore, the thin gold foil should have presented a solid barrier of atoms to the alpha particles, causing them to be stopped.
However, that was not the case. To his astonishment, Rutherford found out that almost all of the alpha particles went through the gold foil as if it was not there. A few particles, though, were diverted, sometimes by as much as 180 degrees. From these results, Rutherford drew the conclusion that most of the atom must consist of vacuum, where the particles could pass freely. The mass of the atom must be concentrated in a tiny nucleus in the center. The alpha particles that were diverted had gotten so close to a nucleus that they had been repelled by it's positive charge (alpha particles have positive charge as well). Thus he posited the following model of the atom:
Atoms have an internal structure similar to the structure of our solar system: In the center there is the nucleus - the "sun", which is orbited by electrons - "planets". This miniature solar system is about 10^-9 metres across, and the nucleus has a diameter of only about 10^-14 m, a hundred thousand times less. The rest is vacuum, except for the electrons, which are far smaller than the nucleus.
The greatest difference to a solar system, besides size, is that the force keeping the electrons on their orbits is not gravity, but rather magnetism. The electrons have a negative charge (like the negative pole of a magnet), while the nucleus is positively charged. The electrons are magnetically attracted to the nucleus, just like the planets are gravitationally attracted to the sun.
Rutherford's atomic model has been superseded by the Bohr model of the atom.
Gold Foil / Target
| \______ Alpha Particles |