How is a massless particle possible? If something exists, must it have mass?
There are various ways of addressing this question. Perhaps the simplest approach is as follows: Physics is an experimental science. It is an experimental fact that some elementary particles - in fact, two that we know of, namely the photon and the gluon do not have (proper) mass. (A third massless particle is the graviton, but its existence was never confirmed). Though a more accurate statement would be that their measured mass is consistent with zero.
By "proper" mass, it means a mass that is the same in any reference frame. Massless particles do react to gravity, and do have energy, though their energy could be different for observers in different reference frames. Thus, one can define an "effective mass", also known as "relativistic mass" for them if their energy is known, using Einstein's famous formula, meff = E / c2. But this effective mass depends on the frame of reference in which we look at them. In different frames, particles (including photons) have different measured kinetic energy, and therefore different masses. If we want to measure the photon "proper" mass, we have to go to a frame in which its kinetic energy is zero; this is the rest frame of the photon (namely, the reference frame in which the photon is at rest). Only in a frame in which the photon is at rest, its kinetic energy would be equal to zero, and so would be its relativistic mass. But in order to do that, we have to be in a frame which moves with the photon. However, the photon travels at the speed of light, which is the same in all frames... So the unavoidable conclusion is that a zero proper mass means that massless particles can in fact never be at rest! Photons are always moving at a constant speed, which is the speed of light.
So experimentally, photons do not have mass. Yet, they clearly exist! And the fact that you can read this answer means they do. So apparently, a proper mass is not a requirement for something to exist. It is, though, a requirement for something to be able to be at rest.
At a more deeper level, the question of mass - and in particular, what causes the mass of the various elementary particles is still a mystery not fully resolved. Within the "standard model" of particle physics, which by itself relies on the framework of quantum field theory, the universe is filled with fields. For each elementary particle there is an associated field: there is an "electron field", a "down quark field", etc. When a field is excited (very much like waves in an ocean) the excited field behaves like a particle. The fields can interact with each other in various ways. Interaction with one particular field - known as the Higgs field (with associated "Higgs particle") generates mass. While some fields strongly interact with the Higgs field (and therefore their associated particles are heavy), other interact only weakly with this field. The photon and gluon fields do not interact at all with the Higgs field, and thus the photon and the gluon do not have mass. But exactly why it is so, is still unknown.