To get at the heart of this question, you first need to imagine what nature would be like if matter were NOT made of atoms. Think a minute about the options ... if there were no atoms, then what would matter be made of? In other words, water would still be water, but there wouldn't be water molecules, so what would the structure of water be?
You may know that the concept of an "atom" or "iota" of matter was originally proposed by a Greek philosopher in the 5th century B.C.! The term "atom" signifies the smallest unit of matter which maintains the identity of the whole.
So ... if you have a chunk of carbon and you divide in half, you have two smaller pieces of carbon. If you take these and divide them in half, you have two still smaller pieces of carbon. If you continue this process, the pieces continue to be carbon, just smaller and smaller. So how long can you keep this divide-in-half-and-still-end-up-with-carbon process?
The answer to this question depends on whether matter is really made of indivisible atoms or the alternative, divisible non-atoms! The alternative would be a lot like what we experience in everyday life, smooth, continuous stuff. No matter how many times you cut butter in half, it is still butter. However, if butter is really made of olein, palmitin, and stearin, at some point - albeit VERY small - cutting the butter in half will destroy its "butter-ness."
So how does the difference between atoms and non-atoms manifest itself on our level? Try the following very simple experiment:
Put a small amount of fine dust - I prefer chalk dust myself - in some water and stir it up. Next, put a drop on a microscope slide and put a cover slip on top, then look very carefully at the chalk dust particles. Note how different particles move. (If you have a small test tube and a bright light source, you can also look at smoke particles in the air inside the test tube).
Robert Brown did this experiment and published his results in 1828. He put microscopic particles of all kinds in water - live pollen, dead pollen, organic dust, inorganic dust - and found the same motion you see with chalk dust particles. The motion, dubbed "Brownian motion," is ubiquitous; everything's doing it!
Do you see why this implies atoms now! No? Think again about the alternative, non-atoms; how would a chalk dust particle move in a sea of "non-atoms," i.e., a smear of water smoothy stuff?
To solidify your ideas, check out the following:
The following simulation shows a single small particle - like a very tiny piece of chalk dust - floating in some fluid like water. I call this particle a "mite." The individual black dots represent the atoms of water while the circle represents the mite, the single particle of dust.
The atoms do not collide with each other (an "OK" simplification for this case), but do collide with the mite. When they collide with the mite, both momentum and energy are conserved. The simulation is just that, a simulation of the real thing. Note the status window at the bottom of your browser, it tells you about the mass of the "mite" relative to the mass of the atoms, the number of atoms, the temperature, and the velocity of the mite. The Toggle Mode button allows you to see or not see the atoms (an option you don't have in the real experiment).