To illustrate the many manifestations of this idea, let's look at some examples:

- Physics: Water boils at 212 degrees Fahrenheit. This fact is so commonplace that it's easy to forget how fundamentally surprising it is. Temperature is basically a measure of how "jittery" the molecules in a substance are. Most of the time, if you increase water's temperature by a degree or two, you make the individual molecules buzz around faster but the liquid itself ("the system") retains all of its basic properties. But at the magical point of 212 degrees, a slight change in jitteriness radically changes the system's behavior. At the critical point, the slight change is just enough to overcome certain forces holding the molecules together, and off they go.
- Computer Science: Say you give a computer a randomly selected problem from a certain class of problems (like finding the shortest route between two points on a road map), and see how long the computer takes to solve it. Of course, there are many ways of "randomly" choosing a problem, so let's say you have some parameters which tell you how likely some problems are versus others. For the most part, a small change in the parameters won't change the complexity of the problem much, but at some critical values, a small change can make the problem much simpler or much more difficult. (For a technical exposition see here.) Papadimitriou claimed that, in some mathematical sense, these are the same kind of phase transitions as in solids and liquids, but I don't know the details on that claim.
- Mathematics: There are several mathematical phenomena that behave like phase transitions, but I'll focus on bifurcations. A dynamical system in mathematics is a system that evolves from one state to another via some rule. Change the rule a little and you'll change the system's behavior, usually not by much, but sometimes by a whole lot. For instance, the system might shift from being in equilibrium to alternating between two states. Change the rules a bit more and it could start cycling though four states, then eight. Another small change could land you in chaos, in which predicting the future behavior of the system is next to impossible.
- Ecology: Okay enough with theory, let's look at some situations where phase transitions matter in a huge way. Ecosystems are adaptive, meaning that they can absorb a certain amount of change while maintaing their basic state. However, Folke et. al. have extensively documented what they call "regime shifts" in ecosystems--changes in ecosystems from one stable state to anoher, very different stable state (think rainforest to desert.) Often these shifts appear triggered by human behavior. Folke et. al. also review ways to increase ecosystem resilience (i.e. make them less susceptible to regime shifts) by, for example, promoting and maintaining biodiversity.
- Economics: Well for starters, there was the Great Phase Transition of 1929, or the current phase transition triggered by sub-prime lending. In both events, small crashes cascaded into much larger ones because of underlying problems in the market: in the first case people buying stocks with borrowed money; in the second case investment in risky mortgages that only make sense when interest rates are low. These underlying problems created a situation where a single "spark" could bring the whole market down.
- Social Sciences: The idea of a "tipping point" actually belonged to sociological theory before Malcolm Gladwell popularized it. It refers to any process that, upon gaining critical momentum, cascades dramatically. The term was first coined to describe white flight: once a critical number of nonwhites moved into a neighborhood, all the whites would head for the 'burbs. It has since been used to describe all manner of trends and fads, as well as contagious diseases. Trends that never catch enough initial supporters will die out quickly, but beyond a certain point, they're unstoppable. "Facebook" ustoppable.

Given their importance and ubiquity, understanding the how, why, and when of phase transitions is a crucial project. The good news is that they're not totally unpredictable--there are certain signs that tell you when a phase transition may be approaching. However, this discussion must wait for another time.