Phase transitions
A phase transition is the transition from one phase of matter to another, which takes place as some external parameter is varied. The everyday examples of different phases are the gas, liquid and solid phase of a given material (like water). The external parameter may be the temperature, the pressure, or the strength of an applied magnetic field, among other examples.
An important distinction is between first order ("discontinuous") and second order ("continuous") transitions (there are also higher order transitions, not very important in practice).
Examples of first order transitions are evaporation (liquid->gas) and melting (solid->liquid). In these examples, it is the density which changes abruptly from one phase to the other. In addition, the energy content of the material makes a jump as well, which means one has to supply energy ("latent heat") in order to evaporate (or melt) a certain amount of liquid (or solid). These are general features of first order transitions.
An example of a second order transition is the appearance of spontaneous magnetization in a ferromagnet (like iron), if it is cooled below a certain ("critical") temperature. Here the word "spontaneous" refers to the absence of any external magnetic field (which would magnetize the material even above the critical temperature). This transition from the "paramagnetic state" above the critical temperature into the "ferromagnetic state" below this temperature is continuous: both the magnetization as well as other quantities (like the energy of the material) change continuously. In particular, there is no "latent heat" associated with such a transition.
Other examples of second order transitions include the formation of the superconducting state of a metal upon cooling below a critical temperature, as well as the appearance of superfluidity in the Helium liquids.
There is also a special combination of pressure and temperature ("critical point") at which the transition between liquid and gas becomes a second-order transition (just before the distinction between liquid and gas phase vanishes completely at higher pressures/temperatures). This is associated with the phenomenon of "critical opalescence", i.e. a "milky" appearance of the liquid, due to density fluctuations at all possible wavelengths (including those of visible light).
It is characteristic of second-order phase transitions in general that such "critical fluctuations" (of density, magnetization etc.) appear near a critical point.