Another in my off-and-on series about condensed matter physics concepts.
Everyone has an intuitive grasp of what they mean by "temperature", but for the most part only physicists know the rigorous definition. Temperature, colloquially, is some measure of the energy stored in a system. If two systems having different temperatures are placed "in contact" (so that energy can flow between them via microscopic interactions like atoms vibrating into each other), there is a net flow of energy from the high temperature system to the low temperature system, until the temperatures equilibrate. Fun fact: the nerves in your skin don't actually sense temperature; rather, they sense the flow of thermal energy. Metals below body temperature generally feel cold to the touch because they are very effective at conducting away thermal energy. Plastics at the same temperature feel warmer because they are much worse thermal conductors.
Anyway, temperature is defined more rigorously than this touchy-feely business. Consider a system (e.g., a gas) that has a certain amount of total energy. That system can have many configurations (e.g., positions and momenta of the gas molecules) that all have the same total energy. Often there are so many configurations in play that we keep track of the log of the number of available configurations, which we call (to within a constant that gives us nice units) S. Now, what happens if we give the system a little more total energy? Well, (almost always) this changes the number of available configurations. (In the gas example, now more of the molecules have access to higher momenta, for example.) How many more configurations? A simple assumption here is that the change in E is linearly proportional to the change in S. The proportionality factor is exactly T, the temperature. At low temperatures, a given change in E implies a comparatively large change in the number of available configurations; conversely, at high temperatures, a given change in E really doesn't increase the number of available configurations very much. (I know that someone is going to object to my gas example, since even for a single molecule in a box there are, classically, an infinite number of possible configurations for the molecule even if we only keep track of positions. Don't worry about that too much - we have mathematically solid ways to deal with this.)
Those in the know will already be aware that S is the entropy of the system. The requirement that the total S always increase or stay the same for a closed system ends up implying just the familiar properties of temperature and average energy flow that we know from everyday experience.