An engineer, a mathematician, and a physicist went to the races one Saturday and laid their money down. Commiserating in the bar after the race, the engineer said, "I don’t understand why I lost all my money. I measured all the horses and calculated their strength and mechanical advantage and figured out how fast they could run..."
The physicist interrupted him: "...but you didn’t take individual variations into account. I did a statistical analysis of their previous performances and bet on the horses with the highest probability of winning..."
"...so if you’re so hot why are you broke?" asked the engineer. But before the argument can grow, the mathematician takes out his pipe and they get a glimpse of his well-fattened wallet. Obviously here was a man who knows something about horses. They both demanded to know his secret.
"Well," he says, between puffs on the pipe, "first I assumed all the horses were identical and spherical..."
You’ve probably noticed that the world is extremely complicated, even if you leave out people’s love lives. To understand it we try to figure out patterns, and then see if we can isolate the things that go to make up those patterns. For example, things slide downhill. Sometimes they stop on the way; sometimes they keep on going through your garden and the back of the garage. You can conclude that something pulls them down (more strongly the steeper the slope), and that something slows them down (that varies in strength depending on the surface roughness and how fast the object is going).
The obvious thing to do is set up some experiments that limit the number of things that change. For example, pick rocks the same shape and slide them down a straight slope of uniform texture with no bushes in the way, and see what you learn about the force that pulls them down. Then vary the tilt, and so on, until you have a model for the force. Then you study the effect of the surface, and so on until you have a picture of friction. Then maybe you worry about air resistance...
Isolate, simplify, and model—and then introduce the complications. It works pretty well.
But what do you do when the pattern is irreducibly "almost symmetric?" You have electrons and anti-electrons, electron neutrinos and anti-electron neutrinos—but you also have muons and anti muons and similar neutrinos. And there are tau particles as well. The groups are called "generations" and the particles in each generation behave in very similar ways to their counterparts in the others; similar enough to use the patterns to form a model to describe them. But they aren’t the same, and the different masses have consequences.
Noether showed that conservation laws (like conservation of momentum, energy, angular momentum) could be associated with symmetries. For example, conservation of momentum is associated with the symmetry of displacements in empty space. Move from one point to another in empty space, and it looks exactly the same: symmetry. And it turns out that some symmetries have representations (mathematical models that reflect the properties of the symmetry) in which some of the parts can represent fundamental particles like electrons. Offer a scientist a powerful tool like that and watch him apply it everywhere he can--it unifies symmetries, forces, and particles in a single bundle.
Analyzing interactions of matter and energy using the symmetries of the relevant fields is immensely powerful—very accurate calculations prove it (QED). Working backwards, by noting that there exist what appear to be parts of the representation and inferring the symmetries, has also been powerful (QCD) and surprisingly accurate (when we’ve been able to actually do the extremely complicated calculations).
I say "surprisingly" because one of the assumptions that goes into the theory is that the particles are massless. This isn’t even close to true for quarks, and the only reason we put up with such nonsense is because the theory works so well. In the real world, and if you want full accuracy, you have to include the masses so that the heavy particles aren’t quite so interchangeable with the light ones anymore. The wonderful symmetry in which all the particles are massless is "broken".
It is only "broken" because we started with the theory where everything was nice and symmetric, so something has to "add on" to the theory to make the proper bits different. It is easier to understand the interactions when you study the problem in that order, but it leads to weird jargon. Sorry.
So what makes the difference?
Higgs (actually several people before him as well, including a superconductivity physicist named Anderson) discovered that you could give a particle or quasi-particle what was to all intents and purposes a mass by the way it interacted with a field with special properties, in particular that it has a non-zero vacuum expectation value. Massless particles could acquire masses in a natural way. It is really a little more involved, since particles in families can share the effects in different ways (e.g. the Z boson is heavy and the photon is massless), but that’s the general idea. Different particles interact more or less strongly with this field, and the effect is to give them what we call mass.
Of course this doesn’t explain why an electron has one mass and a muon a different one. They couple to the Higgs field with different strengths, but nothing yet explains the different coupling strengths. But hey, at least they don’t have to be massless, right?
This kind of field demands a particle to go along with it, and people have been looking for it for years to try to verify the theory. And it looks like it has been found, though there are a few oddities that may just be statistical--we should know by Christmas. (There might be more than one type of Higgs—several theories demand that.)
Perhaps I don’t understand it well enough, but the theory still feels a little ad hoc (*), and several years ago I was predicted that there wasn’t any Higgs. Looks like I was wrong.
(*) Especially that non-zero average value for the Higgs field.