Let's begin with a journey through time (and space, if you live any distance from Cambridge). The year is 1909, nearly seventy years since the death of John Dalton, the physicist who pioneered atomic theory. It's a little over ten years since Sir Joseph John Thomson (J.J. Thomson—not to be confused with the other J.J. Thomson, who was a philosopher) discovered the electron, and created the "plum pudding" model of an atom. This model stated that atoms were a more or less homogeneous mix of positively-charged and negatively-charged particles (protons and neutrons respectively). Electrons were the negatively-charged ones, because Ben Franklin said so in the eighteenth century (simplification, and possibly slightly farcical). Atoms were more of less the coolest kid in town when it came to physics, but scientists still didn't really know all that much about them—the "plum-pudding" model was mostly a wild guess.
Enter our protagonists: Hans Geiger and Ernest Marsden. In Ernest Rutherford's laboratory in Cambridge (Ernest was just a great name to have then—there was a tremendous importance placed upon having that name) they used a radioactive isotope to shoot alpha particles at a really, really, really thin sheet of gold foil. Gold foil, because the aluminium foil we have all come to know and love didn't exist yet, as no one had come up with a way of making aluminium cheap and malleable (gold, while not cheap, was certainly malleable). Alpha particles are the heaviest form of radiation, and are basically just helium nuclei—a dense, positively-charged mass that travels very short distances through even something as diffuse as air (the other two forms of radiation are Beta particles, which are just electrons (and will go several meters in air), and gamma radiation, which is in electromagnetic spectrum). Gold is also very dense, and, subsequently, Geiger and Marsden noticed that it seemed like not all the particles were making it through the gold.
This, however, was not news. Anyone knows that if you shoot particles of something at something else, that something else will likely stop them. With things as small as subatomic particles, many will probably also make it though. If you were to throw sand at a screen door, some will be stopped, even though all could technically make it through. What was news to everyone was that the alpha particles were emerging from the foil at all sorts of odd angles, and some were even coming almost right back at the source.
I'd like to take a breif interlude to discuss computers. Specifically, the lack of them in 1909. This meant that there was no computerized data processing, and no nice automated detectors for alpha particles. If you were to do this experiment in a lab today (as I have), you could just set it all up and then walk away, relying on the computer to record all the data for analysis later (and also possibly put up a sign warning that you're doing an experiment with radioactive materials, as that is the courteous thing to do). Geiger and Marsden had to sit and stare, for hours and hours on end, at a Zinc-Sulfide screen, and watch for the tiny, tiny blips that occured when an alpha particle struck it. Physics experiments were so very exciting back in the day.
Anyway, this was crazy. Absolutely bonkers. Rutherford quickly realized that there was no way that the particles could have been deflected as much as they were without some kind of force acting upon them. The only culprit for such a force would have to be the coulomb force—the force that repels two like-charged bodies (and attracts oppositely-charged bodies), such as the positively-charged alpha particles and the positively-charged protons in the gold atoms. But the only way this deflection would make sense would be if the positive charges in the atom were clustered together, not haphazardly distributed like in the Thomson "plum-pudding" model of the atom.
This revelation led to a revolution in atomic physics, which eventually coalesced into the Bohr model of the atom (which is also wrong, but in a not-quite-as-wrong-and-more-of-a-gross-simplification-and-approximation way), which is the classic atomic shape you're familiar with: the nucleus in the center with electrons in increasingly distant orbits (or shells). This two-part model of the atom was what came out of the Rutherford experiment, which showed that the atom was composed of an inner nucleus (dense, heavy, and positively charged), and an electron cloud (99% of the atom by volume, but almost no mass. Just a few tiny electrons whizzing about, doing their thing, and occasionally acting really weird and confusing quantum physicists). This understanding is part of what has led to many of our important technological advancements (including the transistor, without which you wouldn't be able to be reading this on a computer smaller than a decent-sized room). Also, because it's just plain awesome.