Short answers, pulled from the story.
A gauge boson is a bosonic elementary particle that acts as the force carrier for elementary fermions. Particles described by gauge theory interact by exchanging gauge bosons, typically as virtual particles.
The Standard Model recognizes four kinds of gauge bosons: photons, which carry the electromagnetic interaction; W and Z bosons, which carry the weak interaction; and gluons, which carry the strong interaction.
W and Z bosons gain mass through the Higgs mechanism. The Higgs field undergoes spontaneous symmetry breaking, permeating the universe with a non-zero vacuum expectation value that couples to three of the four electroweak gauge bosons, giving them mass while leaving the photon massless.
There are eight gluons. This number comes directly from quantum chromodynamics, where the gauge group SU(3) has eight generators, and the number of gauge bosons equals the number of generators of the gauge field.
X and Y bosons are hypothetical gauge bosons predicted by the Georgi-Glashow grand unification model. They would mediate interactions between quarks and leptons, violating conservation of baryon number and causing proton decay; searches at facilities including Super-Kamiokande have found no evidence of them.
It is unknown whether the graviton, the hypothetical carrier of gravity, would be a gauge boson. No experimental evidence for the graviton exists, and there is no mathematically coherent theory of quantum gravity; general relativity does have a symmetry analogous to gauge invariance called diffeomorphism invariance.