One series of experiments, known as the Geiger-Marsden experiments (or Rutherford gold foil experiment), was crucial in understanding something so basic that it lies at the very basis of matter itself: atoms. You would think that by the 20th century, scientists must have figured that out already, given the fact that the concept appeared in the ancient cultures of Greece and India.
However, there are many things like this we do not yet fully understand. That is because only with time can we open our minds to the level needed to change perspectives and use the new technology and mathematics that arise. Hence, trouble appears exactly with the most basic concepts, which are the hardest for us to change our views about.
So these Geiger-Marsden experiments proved something crucial. To ‘prove’ something actually means to have an idea of what you want to ‘prove,’ to have a theory, which is not the case here, so let’s say that they ‘discovered’ something new: that atoms have nuclei, and that it is positively charged, where most of the atom’s mass is concentrated. In his first formulation, this nucleus (which was not called ‘nucleus’ until years later) was defined as a “volume of electric charge that is very small and intense,” even considered in calculations as a point charge.

Where did it start and who started it?
Common sense tells us that the ‘stars’ of this achievement were some guys (yes, I agree with feminism to some degree, but scientists were mostly men at the beginning of the 20th century), with the names of Geiger and Marsden. This is partially true. Actually, if you look it up, you will find that in most places, this exact thing we are talking about is called the “Rutherford gold foil experiment,” wrongly so because it was not a single experiment conducted, but rightly because Ernest Rutherford was a significant figure in the story. He was the theorist behind it all and the director of the experiments, while Geiger and Marsden were the actual experimentalists.
Trying to explain some properties of the atom, namely that atoms have no net electric charge while electrons are negatively charged, J.J. Thomson, a British scientist, proposed a model of the atom known as the ‘plum pudding model.’ No joke. And it actually means the atom is like a plum pudding. For reference, here you have a picture of one:

What does that actually mean, leaving food aside? Thomson believed that an atom must be a spherical volume of positive charge, with electrons spread like raisins throughout the sphere, hence the lack of charge for the atom.
The experiments
Ernest Rutherford worked at the University of Manchester when Hans Geiger visited him in 1906 to discuss physics problems. Impressed by Geiger’s skills and ideas, Rutherford invited Geiger to come and work with him some more, and Geiger also brought one of his students, Ernest Marsden.

Experiments were conducted in 1908, 1909, 1910, and 1913, with the most important probably being the 1909 one. Then, they proposed a simple experiment where a beam of alpha particles is sent through some material to see how the beam behaves. Alpha particles are positively charged particles, consisting of two protons and two neutrons bound together. If Thomson’s model was right, the beam should be deflected at a small angle, at maximum. However, the most common result would have been straightly going through the material, which was actually a foil.
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When trying with different elements, they observed that metals with a higher atomic mass, such as gold, reflected more alpha-particles than lighter metals such as aluminum, so they kept on experimenting with gold.
It was absolutely incredible to see that not only were particles being deflected at large angles, but some particles even came back in the direction of the beam! That is analogous, with the perspective scientists had back then, with shooting at a paper and seeing the bullet coming back at you. The Thomson model advocated for an almost empty volume of charge when this clearly showed that something really massive must be in there somewhere. Then, of course, Rutherford figured out that it must be the atom’s nucleus, but it completely changed how we looked at atoms!
Geiger further explored this experimentally and came up with some conclusions that show us better why gold is the preferred element, such as that the most probable angle of deflection increases with the thickness of the material or that the most probable angle of deflection is proportional to the atomic mass of the substance. Gold is easily made into a very thin foil, a very, very, very thin, with the thickness of almost an atom. For the experiment to go as expected, it doesn’t have to be by any means disturbed by such things as a too-thick foil.

“It was quite the most incredible event that has ever happened to me in my life. It was almost as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you. On consideration, I realized that this scattering backward must be the result of a single collision, and when I made calculations I saw that it was impossible to get anything of that order of magnitude unless you took a system in which the greater part of the mass of the atom was concentrated in a minute nucleus. It was then that I had the idea of an atom with a minute massive centre, carrying a charge.”
Ernest Rutherford
Is this the way we look at atoms now?
In some ways, yes. Firstly, the Rutherford model completely does not respect Newtonian mechanics, which was troubling at first and would have been even more troubling if quantum mechanics hadn’t been on the way. Accelerating charged particles makes them radiate energy, making electrons spiral into the nucleus, losing energy, which is clearly not the case.
So scientists had to incorporate quantum mechanics into the game, and the Rutherford model (where electrons kind of orbit the nucleus in the same way planets and satellites do) was replaced with an ‘improved’ model of the atom, the Bohr model (which is actually called the Rutherford-Bohr model). We now know that a quantum interpretation is essential. Electrons don’t actually orbit the nucleus just like planets, in clear orbits (as shown in most textbooks and representations), but rather in a sort of cloud.
…because gold is one of the most malleable metals, with a high atomic number and very dense. Besides, gold doesn’t oxidize, so oxygen wouldn’t be part of the layer or affect the results.