This explanation is provided by the team of Stephen Reucroft and John D. Swain, professors at the Department of Physics of Northeastern University.
First it's probably a good idea to review of what an atom is made of. Basically, it contains a nucleus, holding some number (call it N) of positively charged protons, which is surrounded by a cloud (N) of negatively charged electrons. The force that holds the electrons and protons together is the electromagnetic force. The number N tells you what element you have: for hydrogen N equals 1, for helium, 2, and so on.
The same electromagnetic force that draws opposite charged electrons and protons together tries to push the protons (which all have the same charge) away from each other. To avoid this separation, another particle comes into play in the nucleus: the neutron. Much like a proton in mass but without electric charge, the neutron is essential for holding the nucleus together. At short distances (i.e. within the nucleus), a very strong force, more powerful than electromagnetism, takes over and attracts the protons and neutrons. For most elements, there are several possibilities as to how many neutrons can fit into the nucleus, and each choice corresponds to a different isotope of that element.
Suppose you want to pull an atom apart. The first thing you need to do is get rid of the electrons. There are lots of ways to do this. You can shine light on the atom, or expose it to another form of electromagnetic radiation having an even shorter wavelength. Also, you can hit it with particles such as electrons or other atoms. Actually, light is made of little chunks called photons, so shining light on an atom is really just a special case of whacking it with other particles.
Heat will do the trick too, but indirectly. It makes atoms move quickly and hit each other. The first electron will come off fairly easily, leaving an object with a net positive charge (called an ion). Each successive electron tends to be harder to strip, as it sees itself as part of an object with an increasingly greater positive charge.
After all the electrons are gone, you're left with nothing but a nucleus. Because the strong force holding the protons and neutrons together is stronger than the electromagnetic one, knocking the nucleus apart into pieces demands more energy than removing the electrons. Even so, the principle is the same: hit it either with photons (but now with those having much more energy than visible light photons) or any of the enormous zoo of particles discovered by high energy physicists. Neutrons are particularly useful because they have no electric charge. Thus, they can cruise straight into a nucleus, unimpeded by electromagnetic forces. Typically it takes about a million times as much energy to get stuff out of a nucleus as it does to strip an electron from an atom.
Although we've talked about breaking an atom apart in steps, you can, of course, hit a complete atom (electrons and nucleus) with something; if you hit it hard enough, you'll get a load of bits and pieces.
Two more points are probably worth covering while we're at it. The first is that if you hit protons and neutrons hard enough, you find that they in turn are made of even smaller pieces, called quarks. Quarks are held together by the same strong force that holds the nucleus together (though the details of how it works in the two cases are a little different). So far, we have no evidence that electrons have anything smaller rattling around inside them.
The second point is that if you hit things at very high energies, you don't just get pieces, but you also make brand new particles that weren't there before! The theoretical framework to describe this process is called quantum field theory, and the field of physics that specializes in the looking at the creation and destruction of new particles is called high-energy physics. The hope is that by looking deeply enough inside matter--by pulling it apart and making new forms of it--we will better understand the machinery behind the universe we see every day.