How Do We Know that Atoms Exist?
Atoms were first considered as the smallest element of matter. What happens if you took something and cut it in half, then cut that in half and kept going? The original concept of an atom follows from a presumed result of this thought experiment. Eventually you wound up with something that couldn't be cut. In fact, the word atom comes from the Greek word for uncuttable.
Of course turning this thought experiment into a real one is not practical, so the existence of atoms was debated for centuries. The systematic study of chemistry allowed John Dalton formulate the modern theory of atoms. It is based on the observation that different materials react with each other in fixed mass ratios. For example, if you chemically combine hydrogen and oxygen to form water you will always find that the mass of the oxygen used is 8 times the mass of the hydrogen. It was found that all chemical reactions behave this way. That is how all of the known chemical substances, made of molecules, could be constructed from a much smaller number of atoms. If matter consisted of tiny particles that would only combine in certain ways then this is the way things must be.
But that doesn't preclude other possibilities. The same experimental results would also be found if there was just some peculiarity in the way chemistry worked that forced this behavior. So what other evidence is there?
The study of gasses showed that there was a relationship between the temperature, mass, volume and pressure of all gasses. The Kinetic Theory of gasses states that gasses are made of identical particles that are moving at high speed and they bounce off of each other and the walls of their container. The recoil of these bounces is the source of pressure and speed is determined by the mass and the temperature. From these assumptions the behavior of gasses can be predicted. So here we have another, very different phenomenon, that is explained by the assumptions that matter is made of tiny bits.
This evidence was enough for most people, but not everyone. They thought that atoms and molecules were just a convenient fiction that helped get the right answer, but had no physical reality. This isn't as silly as it might sound. Such convenient fictions do exist. An example of this are magnetic lines of force. Magnetic forces act along these lines and they are a convenient way to compute many effects in electromagnetism. But they don't actually exist.
The clincher came from a well known person, but most people are surprised to learn that it was Albert Einstein. In one the four incredible papers he published in 1905 he showed that Brownian motion could be explained by assuming that matter was composed of tiny particles. His paper not only predicted what the motion should look like but allowed for the computation of the mass of a single molecule. At this point no scientist seriously doubted the existence of atoms. There were just too many different kinds of phenomena that were explained by Atomic Theory both qualitatively and quantitatively.
But now we have even more evidence. Here's a few examples:
The quantum mechanical description of atoms predicts, to an astonishingly accuracy, the details of atomic spectra.
Individual ions (electrically charged atoms) can be manipulated with electric and magnetic fields.
There are even elevation maps of surfaces that show the bumps caused by individual atoms.
This technology has even been used to model the IBM logo with individual atoms!