Phase transition is the technical term used in physics to describe the phenomena such as the melting/freezing of ice/water, or the boiling of a liquid to make a vapour. In equilibrium just below the temperature at which ice melts, it is a solid, and in most senses of the word, there is no liquid present at all. In equilibrium just above the temperature at which ice melts, it is a liquid, and in there is no solid present at all. Raising the temperature by just a few milli-degrees is sufficient to transform completely the properties of water-substance. Transitions such as this are called first-order for technical reasons.
Not all phase transitions proceed in this way. For some transitions there is a critical temperature below which some kind of structure or order begins to appear. That is, just below the critical temperature TC the structure or order is not fully developed, but in some sense 'grows stronger' as one proceeds below TC. Thus both phases co-exist just below TC with the low-temperature phase grwoing stronger the further the temperature is lowered below TC. These phase transitions used to be called second-order, but are now more properly known as continuous.
The most common example of a continuous phase transition is the liquid/vapour system. Consider a volume filled with vapour at constant pressure. This might correspond to vapour trapped in a very loose balloon. For the sake of definiteness let's assume that we have water vapour and that there is no liquid water present in the container. As one lowers the temperature, the density of the vapour increases because from the perfect gas law the molar density n/V =P/RT. Eventually one reaches a point where some liquid condenses. Both liquid and vapour coexist, but the amount of liquid increases as the temperature is lowered below this condensation temperature. But even well below TC, there is still vapour present above the surface of the liquid.
Above TC for a continuous phase transition, the substance is constantly fluctuating into a state which is close to the new state which will take over at lower temperature. These fluctuations are generally confined to just a few molecules at a time and take place on the length scale of a few nanometres. However as one gets closer and closer to the transition temperature, the stability of the two phases (Technically: their specific Gibbs Free Energy) becomes closer and closer and the random fluctuations grow larger both in their physical extent and the length of time over which they last. Eventually the length scale and the time scale become so large that the system simply fluctuates into the new phase. TC marks the temperature at which length scale and timescale of the fluctuations diverges.
Critical Temperature in a Liquid/Vapour System
To appreciate what the critical temperature indicates in a liquid vapour system, consider the case now of liquid water heated in a closed container. As the temperature of the container increases, the density of the water falls (slowly) and the density of the vapour rises (rapidly). If the volume of the container is chosen so that all the liquid does not evaporate, at some point the densities of the liquid and vapour become equal. For most fluids this occurs when the liquid density is around one third its "normal" value. For water this corresponds to a pressure of around 220 atmospheres and a temperature of 647.3 kelvin (374.2 celsius).
647.3 kelvin corresponds to the critical temperature of water. At this point there is no distinction between the dense vapour and the low density "liquid" and so there is no latent heat associated with the transition. Both are just matter in a fluid state. Near this point there is only a small distinction between the between the dense vapour and the low density "liquid" and the latent heat grows smaller and smaller as one approaches TC.
Opalescence in a Liquid/Vapour System
On a microscopic scale, the molecular randomness cause a liquid near TC to constantly fluctuate into nearly vapour-like volumes and back again. As one approaches TC the length scale of the fluctuations grows. This is because the energy required for a fluctuation into the vapour state becomes smaller as one approaches TC. Eventually the fluctuations occur on the scale of a fraction of a micron: i.e. of the same order as the wavelength of light. If there is a difference in refractive index of the vapour and liquid phases (and there is generally a small difference) then light will be strongly scattered and the mixture of the phases appears cloudy. This phenomenon is known as critical opalescence.
However the critical points of all practical substances occur at pressures of above 10 atmospheres and so require special safety precautions to observe. As mentioned above, the pressure required for water is greater than 200 atmospheres. The combination of high pressure and glass windows is generally a troubling one.
Opalescence in a Binary Liquid System
Some binary fluid mixtures also show a critical temperature. Above TC the fluids are miscible, but below TC they separate into two separate phases. As one cools a binary fluid towards TC from above, the fully mixed phase is constantly fluctuating into phase-separated volumes and back again. As one approaches TC the length scale of the fluctuations grows and eventually reaches the scale of a fraction of a micron: i.e. of the same order as the wavelength of light. As with the liquid vapour systems, light will be strongly scattered and the mixture of the phases appears cloudy. This critical opalescence is exactly analogous to that seen in liquid/vapour systems.
One can imagine that one of the fluids (e.g. in our case the hexane) is a vacuum. The methanol "evaporates" into the vacuum. Since the volume of the hexane is fixed, as the temperature increases, the density of the methanol "evaporated" into the hexane equals the density of the methanol "liquid". This marks the critical point of this system. The analogy is complicated because simultaneously the hexane is "evaporating" into the "vacuum" of the methanol, but the basic analogy between the binary fluid system and the liquid/vapour system is sound. Thus binary fluids allow one to observe critical opalescence, with negligible safety risks. And of course, observing it on the web your safety risk is reduced still further.