What is the simplest possible experiment that proves that there are atoms? As I known all the matter consists of atoms. What is the simplest possible experiment (with as little scientific equipment as possible) that can prove that there are atoms?
 A: To show kids atoms in action I use Brownian Motion similar to what Robert Brown did with his pollen grains. You need a microscope with at least X400 magnification, some whole milk, a glass slide and cover slip. Place a drop of milk on the slide and a drop of water. Cover the milk-water with a cover slip and view under the microscope. For a whole class it's good to have a camera and projector on the microscope.
What you will see are small fat globules bouncing around. The motion is random walk caused by random bombardment of the water molecules against the globules. The dynamics of this motion were predicted by Einstein in 1905 as a function of viscosity, particle size and temperature. Given those known quantities, the dynamic model and the mean size of the random walk gives an estimate of the size of the atoms (or molecules in this case). 
Einstein's work was a turning point in the belief that atoms actually exist. Before that time scientists were divided.
A: Chemistry. The things we call "atoms" are chemical objects, different of the atoms of the ancients (closer to "elementary particles"). The existence of chemical atoms implies integer proportions in chemical reactions.
I'd vote for electrolysis of water. The volume of $H_2$ and $O_2$ increases in integer proportion 2:1.
EDIT: from discussion in the comments, it seems that a single proportion is not enough to settle the question, in this sense stoichiometry is not the "simplest" possible, as a whole corpus of proportions is needed to show that some basic multiple is going on. EDIT2: note also that weighting the products of electrolysis you could have a first hint of nucleon mass. Again, lot of measures needed to be sure, isotopes and all that.
A: Oil drop on liquid is about the simplest I think, - take a bowl of still water - dust it with some fine powder - put a small drop of oil of top - The oil spreads out over the surface and generally forms a circle the size of which you can see from the dusting. The thickness of the drop on the liquid can be calculated if you can estimate the volume of your initial oil drop. The thickness should come out at about 1 Anstrom - roughly 1 atom thick. 
I saw this demonstrated in School (US High School) as a demonstration of atomic nature of matter. 
A: Brownian motion would be considered pretty good evidence of atoms. Originally, Brown observed pollen grains moving in water in ways that could not be explained otherwise. 78 years later, Einstein came up with a theoretical model of Brownian motion1 that was later borne out by experiments by Jean Perrin (see this). A translation of Einstein's paper is available here.2
You could reproduce it easily with other small particles suspended in a different liquid (though water might be the best). You would need to make sure that the liquid is not disturbed by outside forces, and you would of course want to try other substances to show that it is a general phenomenon.
Note: It looks like this was explained better here, which is absolutely worth reading.

1 Marian Smoluchowski also did work on Brownian motion.
2 A note on Brownian motion: DanielSank pointed out in a comment that the Langevin equation, one of the most important parts of the mathematical model behind Brownian motion, does not necessarily imply the existence of atoms or any particles. However, Einstein's use of molecular-kinetic theory does imply the existence of atoms.
A: About the question ,(case 1)Is it that You actually believe that matter is made of atoms in theory and just want to cross check with a simple practical experiment? or (case 2) You actually doubt the existence of atoms itself and want to check it out with an experimental proof.
(case 1) Take a flask/container with an attached barometer and fill it with a mole(2 grams) of hydrogen gas . Let the temperature maintained be T. Let the pressure reading by the barometer be P. Now at the same temperature T remove the hydrogen gas completely and fill it with a mole of nitrogen gas(28 grams) and you find the pressure reading P. Hence the number of molecules in 2 grams of hydrogen gas is same as that in 28 grams of nitrogen. 
Because according the ideal gas law PV=nRT where P is the Pressure , V is the Volume ,T is the Temperature , R is the gas constant and n is the number of molecules, since P,V and T were same n is also same. Theoretically the mass of hydrogen molecule is 2 atomic mass unit(amu) or 2*1.6605*(10^-24) grams. Then how many number of molecules constitute 2 grams of hydrogen gas each with a mass of 2 amu. Just by doing math we get 6.023*(10^23)(avagadro number) molecules count. Now same number of molecules of nitrogen are present corresponding to 28 grams which would give the molecular mass of 28 amu for nitrogen molecule which is theoretically correct. 
(case 2) In case you doubt the existence of atom itself then you could consider Rutherford’s gold foil experiment where a thin gold foil is bombarded with a beam of alpha radiation. With out going to the deeper details of the experiment the scattering of some alpha rays in different directions and some being able to pass right through, gives an abstract view that matter consists of smaller constituents. Of course the experiment was actually conducted to know the structure of atom and that existence of nuclei was an outcome of the experiment but the scattering of alpha rays just pushes us to the thought that the matter is made of some smaller constituents. The Brownian motion also gives a notion of atoms and molecules.   
