What is a quantum spin liquid?
That's a surprisingly controversial question! The term was originally coined in the context of some theoretical toy models, but when you try to apply it to the complexities of real materials people start disagreeing pretty quickly about what the precise definition should be. I think a reasonable starting point for a modern definition is "a topologically ordered phase arising from frustrated spins," but not everyone even agrees with that.
So, to elaborate, there are three "traditional" magnetic phases a material can be in: paramagnetic, ferromagnetic, and antiferromagnetic. Paramagnetic phases are highly disordered; in the absence of an external field, the spins are more or less randomly oriented due to thermal fluctuations. This happens to all magnets at high enough temperatures and is analogous to a gaseous phase. In some materials, as you cool them down the intrinsic tendency of the spins to align with those nearby will win out over the jiggle and the spins will all "freeze" in a single shared direction. This is ferromagnetism and it's a highly ordered phase, analogous to a solid. More rarely, you'll have a material where the spins want to point the opposite direction of those nearby instead, leading to an antiferromagnetic phase at low temperatures, where the spins alternate direction as you move through the material. Like ferromagnetism this is also a highly ordered phase, just a different one, similar to how there are lots of different forms of solid ice. (Which one you get depends on whether the exchange interaction or the magnetic dipole interaction is dominant in the given material.)
But there are subtleties to antiferromagnetism that ferromagnetism doesn't have. I said that antiferromagnets alternate spin directions between adjacent atoms, but imagine a triangular lattice. The geometry of the lattice simply doesn't allow configurations where all adjacent spins are antiparallel, some will always be parallel. We call this situation geometric frustration.
Now, in the case of an infinite triangular lattice it turns out that there's a unique configuration called the Neel order that minimizes the conflict between adjacent pairs, so you get basically normal antiferromagnetic behavior—at low temperatures, the spins pick an orientation and freeze into a rigid, "solid" formation with spins as antiparallel as they can be. But that's not always the case. There are other frustrated geometries, particularly the Kagome lattice, where the lowest-energy configuration is highly underconstrained. Each local patch comes with multiple equally favored configurations, and these are independent; your choice in one patch doesn't fix the configuration for the rest of the material like it does with the triangular lattice. Systems like this won't freeze into a highly ordered phase like ferromagnets and antiferromagnets, there's no symmetry breaking, but they also aren't completely disordered like paramagnets. The desire of the spins to antialign still imposes local constraints on the configuration, they just don't fix the global arrangement. Carrying on the solid/gas analogy from before, this kind of short-range order is the hallmark behavior of liquids, so we call these systems spin liquids.
But why "quantum"? In most familiar liquids, the lack of long-range order comes again from thermal fluctuations, but that's not the case here. In a spin liquid, even at zero temperature, geometric frustration ensures that you'll have this local freedom; the extensive (in the technical sense of "proportional to the number of atoms") multiplicity of global configurations represents a genuine degeneracy of the ground state. In a classical setting, all that means is that, at zero temperature, one of the many ground states will be selected at random. (That's sort of what a spin glass is, to answer your question about the relationship between the terms.) But quantum mechanics opens up more exciting possibilities. Rather than just picking one classical ground state, the system can be in any superposition of all the available ground states, and when there are extensively many of them like this that can lead to qualitatively new behavior. In particular, you can get "topologically ordered" states, which possess a web of intricate entanglements between even parts of the system that are spatially distant from one another. As a result, the excitations aren't the obvious local modes (a single electron, a single spin flip) but some long-range collective motion which typically has a fraction of the quantum numbers of the local mode. For instance, in a typical QSL, the smallest excitation looks like flipping a "spin-1/4" particle from down to up or vice versa—which is of course impossible in a literal sense, but nevertheless. This has been one of the major research frontiers of condensed matter physics for the last several decades.
