Characteristically, a critical point occurs somewhere anytime you have a continuous phase transistion. That is, if you have two phases of a substance that themselves share their intrinsic symmetries. The classic example is the critical point associated with the liquid gas transition, as you note. Liquids are isotropic and homogenous, gases are isotropic and homogenous, they only differ in their densities at lower pressures. This suggests that there is some point in phase space where these two phases are smoothly connected. For most materials this corresponds to high pressure. At lower pressures you might have a discontinuous transition from one of these to the other, just like the boiling we observe for the ordinary liquid-gas transition.
Why do we care? Well, critical points are rather special. Since they lie right at the edge of the phase coexistence curve, they have peculiar properties. In particular, they are characterized by a diverging correlation length, meaning the system can have very long scale correlations, that is, very well separated parts of the system can be highly coordinated together. This gives rise to the critical opalescence you mentioned. At the liquid-gas critical point you get very long scale fluctuations in the system, which can get large enough to scatter light, just like the large proteins that scatter light in milk, giving the fluid a sort of milky character.
Fine, but again, why do we care? Well, critical points are very nice for theory. Since the correlation length diverges, the physics of the system becomes independent of any of the microscopic physics. The physics of the system takes on a life of its own, independant of all of the microscopic particulars. This gives rise to all sorts of power law dependencies for things like susceptibilities, heat capacities and the like. It also gives rise to fractal behavior. Both of these things happen because there is no longer a characteristic length scale in the system, all of the physics must be scale free, properties that both power laws and fractals have.
Again, why should we care? Well, since the system's physics takes on this scale free behavior, and a life of its own independent of the microscopic description, we observe a general phenomenon known as universality, very different systems can start to look very much the same. For instance, the liquid-gas transition you mention has the same scale free behavior at its critical point as a very simple model known as the Ising Model does at its critical point. The spreading of disease at its critical point looks like percolation.
Great, so things look the same, again why do we care? Well, some of these models are much simpler than others. The Ising model is much simpler than trying to simulate a "realistic" liquid, but if the behavior is the same at the critical point, we can hope to make both computational and theoretical progress by studying simpler models that share the same universal behavior as much more complicated models, and rest assured that we aren't wasting our time. Those simple models, if designed right, will have exactly the same behavior at the critical point.