The Higgs boson is a theoretical consequence of the Standard Model of physics, a theory which unifies three of the four known fundamental forces in the cosmos: the strong nuclear force (which explains nuclear energy), the electromagnetic force (which explains electricity, magnetism, light, and the basics of chemistry), and the weak force (which explains a particular kind of radioactivity). Fundamental particles predicted by the Standard Model include the electron (familiar as the carrier of electricity) and the photon, which is an irreducibly small particle of light (The existence of photons leads to noise in digital photography). Other predicted particles include the quarks, which make up the familiar protons and neutrons found in the nuclei of atoms. All of the particles predicted by the Standard Model have been discovered — except the Higgs boson. As far as I know, no subatomic particle has ever been found which cannot be explained by the Standard Model.
The Standard Model does not contain an explanation of gravitation: the best current theory is Einstein's General Theory of Relativity; however, unlike the Standard Model, general relativity is not compatible with quantum mechanics. A logical extension of the ordinary classical or Newtonian mechanics on a very small scale, quantum mechanics posits that energy is not infinitely divisible, but rather is bound up in quanta, or small packets. [Back when I was at Caltech, I proposed doing some research in the interaction of relativity and quantum mechanics, but was told not to bother with it, because my proposal threw out lots of the accepted theory of that era.]
The Standard Model was developed in the 1960s and is based on symmetry arguments. From the 1920s through the 1960s, experimenters had discovered a wild variety of subatomic particles, and many despaired of the fundamental idea that nature is ultimately simple. Eventually attempts were made to put these particles in a rational order. In chemistry, the development of the Periodic Table of the Elements immediately led to the prediction of previously unknown elements, which were indeed discovered either in nature or subsequently synthesized. Likewise, the development of the Standard Model led to the prediction of various subatomic particles — and multiple researchers quickly proposed the existence of what is now called the Higgs boson.
Since the Higgs boson has the interesting property of giving mass to the other particles in the Standard Model, it is quite important to our theoretical understanding of the cosmos. The existence of the Higgs boson explains a lot; the non-existence of the Higgs boson means that some other theory, such as a quantum gravity theory, is needed to explain the existence of mass.
The Higgs boson is often referred to in the press as the ‘God particle’, an unfortunate name taken from a popular science book. That name comes from the theory of materialistic monism, a theory that states that the observable cosmos is all that exists; and this monism is at the core of much modern scientific and secularistic thought — even American public school pedagogy is based on materialistic monism.
The rational consideration of existence goes by the name metaphysics. Nowadays that word has an irrational New Age connotation, and this is partially due to Catholic universities denigrating the study of metaphysics in favor of more worldly and materialistic philosophies; nature abhors a vacuum, and so the word metaphysics was taken over by others. However, metaphysics is a good solid subject: it merely asks the questions ‘what exists?’ and ‘what is it like?’ The existence of the Higgs boson is a currently a metaphysical question, as is the question of the validity of the Standard Model.
The standard history of science ends with Democritus and restarts again with Galileo, ignoring everything from Plato through the Medieval schoolmen. But we can still learn things from these ancient thinkers. In particular, we can consider the Catholic notion of the Great Chain of Being, which posits — via an argument of fittingness — a hierarchical structure of all that exists, which starts from God and continues down through inanimate matter.
The Chain of Being ends in fundamental matter, which is quite remote from the Source of all being. With matter being so remote, it is fitting to assume that the properties of fundamental matter are very simple, and get simpler the closer we look. So the wild variety of chemistry devolves to a simpler set of chemical elements, which are made up of simpler, well-ordered particles; these particles are vast in number, but are reduced to a small collection as described by the Standard Model. The existence of the Periodic Table of Elements is not in dispute, while the existence of the Standard Model remains a metaphysical problem.
As we go all the way down the chain, we ought to find simpler models with simpler interactions, which display greater symmetry — models that mathematical physicists call beautiful. Beauty is a great organizing principle in fundamental physics, with the more beautiful models having a greater likelihood of being true. Beauty and simplicity was one of the factors which led to the acceptance of the Standard Model.
As we move up the chain of being, these symmetries are broken, and things get messier. The periodic table of elements gets messier as the elements get heavier, chemicals have more complexity as they get larger, and living things get more complicated as they go up the chain: protozoa are more more complicated than viruses; fungi are more complicated than protozoa; plants are more complicated than fungi; higher animals are more complicated than lower; and rational animals (which as far as we know only consists of Man) are the most complicated beings in existence. Consider the existence of sexuality — what is pretty simple in lower orders becomes highly messy in human beings.
Platonic and Catholic philosophy also posits the existence of a higher order of being beyond the material world — a metaphysical, intellectual, or spiritual world. It is beyond and above mere matter, and is outside of space and time. Rational animals are seen as straddling these two worlds, being both material and simultaneously spiritual.
Part of the order of being in the metaphysical realm is mathematics. According to Plato, an understanding of mathematics is a great step towards enlightenment — for example, being able to see with the mind's eye, a true, mathematically perfect triangle, beyond the mere material drawn copies of triangles. Because math exists outside of space, time, and matter, it is a reliably true guide to understanding the material world; this is implicit in the modern sciences, especially in physics.
In his parables of the divided line and the cave, Plato asks us to go beyond mere opinion to more solid reality, which includes science of the material world, and mathematics in the metaphysical world — and even higher. Sadly, so many people live and die because of mere changeable opinion, with the most opinion-bound and power-hungry people being the worst of humanity. We can break the chains of this bondage to opinion by rising to contemplate higher things.
As we ascend into the higher realms, things get simpler. Angels and demons, having intellect but lacking bodies, are simpler than humans. Mathematics is extremely complicated, but less complicated than human psychology, and the logical foundations of math are much simpler than say, non-linear mathematical analysis. We reach ultimate simplicity when we rise to the Source, or God. Spiritual directors say that living an excessively complicated life is problematic, and that simplifying your life (in the upwards direction) is key to ultimate blessedness.
God is not a particle, but the existence of the Higgs boson is still an important metaphysical question.