In around 460 B.C., did a Greek philosopher, Democritus, developed the idea of atoms. He asked this question: If you break a piece of matter in half, and then break it in half again, how many breaks will you have to make before you can break it no further? Democritus thought that it ended at some point, a smallest possible bit of matter. He called these basic matter particles, atoms. He also believed that atoms differed from each other only in size and shape, and different substances with their distinct qualities were made up of different shapes, arrangements, and positions of atoms. Atoms were in continuous motion in the infinite void and constantly collided with each other. During these collisions they could rebound or stick together because of hooks and barbs on their surfaces. It was thought that atoms were neither created nor destroyed. At this stage, atoms were only a hypothesis, without any experimental evidence to support their existence.

In the 1800's an English chemist, John Dalton performed experiments with various chemicals that showed that matter, indeed, seems to consist of elementary lumpy particles (atoms). Although he did not know about their structure, he knew that the evidence pointed to something fundamental. Dalton figured out the "stoichiometry" of a lot of different compounds, that is the amounts of different elements that they contain, and he showed that compounds are formed from elements in very definite proportions. For example compounds can be formed from the elements carbon and oxygen, but only in whole number ratios, giving us carbon monoxide (CO) carbon dioxide (CO2) among others. This only made sense if separate carbon and oxygen atoms existed. At this stage in history the existence of atoms was upgraded to a theory.

Today we have a huge amount of evidence for the existence of atoms. We even have experimental tools available which allow us to "photograph" atoms directly! For example, take a look at the "quantum corral" of atoms on IBM's website:
http://www.almaden.ibm.com/st/
So the existence of atoms is now a definite fact.

This progression from hypothesis to theory to accepted fact is very common in science as we learn more about a subject through time. It is remarkable in this case that the ancient Greeks got a lot of it right, over 2000 years before there was any substantial experimental evidence!

This is a very long and complicated story.I suggest that you look up a couple of people that were involved. Some of them are James Chadwick (1891-1974), Sir William Crookes, Michael Faraday (1791-1867), Svante Arrhenius (1859-1927), J.J. Thompson (1856-1940), G.J. Stoney (1826-1911), and Earnest Rutherford (1871-1937). As you can see, there were a lot of different people involved and none of them individually found the atom but they found parts of them like the electron, proton, and neutron. All of their work put together helped us find the atom. Even today more things are being discovered about atoms.

This is a great question! For a long time, people have thought that all matter is made up of tiny, indestructible elements (atoms).
But atoms are hard to see. They're much too small to be seen in a microscope, for example. So it was only about a hundred years ago that people had experimental evidence that atoms exist.
Imagine you were watching a soccer game played with a glow-in-the-dark ball, and suddenly the stadium lights went out. Now you can't see anything except for this glowing ball. You wouldn't be able to see the soccer players, but if the ball kept bouncing around back and forth, you'd know the players were still on the playing field.
This is an "indirect" observation of the players, because you don't actually see them there: you only see the effects of their actions. A similar effect led to one of the first, indirect, "observations" of atoms.
If you look at a little bit of water under a microscope, you'll see a lot of things - bacteria and so forth. Some of the things you'll see are little specks of dirt or dust or something like that. These specks seem to move around a lot, even though they don't have any way to propel themselves. They move around randomly, jiggling back and forth and wandering around the water.
The explanation of this motion came from Einstein, who pointed out that if water was made up of atoms, then the atoms would be constantly moving around. Every so often, an atom (or a molecule, which is a clump of atoms) would happen to bump into a dust speck, and when it bumped into the dust speck you would see the dust speck move a little.
When Einstein carefully analyzed the motions that a dust speck would have, he found that it matched the experimental results quite well.
This is one bit of evidence that atoms exist. It's indirect evidence, because you still cannot see the atoms, but you do observe the effects of their motion.