In nature, there are four fundamental interactions/forces: electromagnetic, weak, strong, and gravitational. These forces are known to be carried by particles called gauge bosons. The rules of quantum theory say that gauge bosons are supposed to be massless, but in nature, only three of the four forces—EM, strong, and gravitational—have massless carriers. The weak force, by contrast, has carriers that are actually really heavy, as subatomic particles go. This made physicists frustrated for a while, because they couldn't figure out how to make the theory of the weak force get off the ground.
But theoretical physicists found a detour around this no-mass rule which is now called the Higgs mechanism. They postulated that the electromagnetic and weak forces, at high temperatures/energies such as was found in the early Universe, were unified together in a single massless force called the electroweak force, and something called the Higgs field existed. When the Universe was hot, this Higgs field/electroweak configuration was stable, but eventually the Universe got too cold for it to exist and it froze. When this happened, the electroweak force "gobbled up" most of the Higgs field, which gave some of its force carriers mass: these newly massive force carriers became the carriers for the weak force. The massless remainder of the electroweak force became the carrier for the electromagnetic force, and the "leftovers" of the Higgs field became the Higgs boson.
As it turns out, this creative fix to the theory only works if everything that interacted via the weak force was also massless during this early epoch of the Universe and given any mass it had through the Higgs field. But all matter (that we know about) interacts via the weak force, so all known matter was given mass via the Higgs mechanism. This little story is an essential part of the Standard Model of particle physics.
Until possibly now, the Higgs boson was the only Standard Model particle yet to be detected in collider. Its existence—or absence—defines particle physics as we know it.