Usually the way to increase the temperature of an object is to add energy to it. What form has this energy taken?

One of the possible forms of energy that an object can have is small scale vibrations or motions. This confers kinetic energy to the moving parts, and also elastic energy, because there are temporary local stretches and compressions as the different parts move relative to each other. These motions can be very fast and very small scale and yet still store a lot of energy.

Because the motions are very small scale, we can't directly see them, but this is the energy whose presence is indicated by the measured temperature.

Cooling off and warming up: energy in motion
Because energy is conserved, the only way an object can gain energy is to take it from some other object. We can add thermal energy to an object directly, by placing it in contact with a hotter object. The small motions that represent the thermal energy in the hot object give rise to small motions in the cooler one, which is how energy is transferred. We will then say that we have heated the cooler object, or say that we have added heat to it. "Heat" refers to the increase in thermal energy of the cooler object. Since energy is conserved, we have removed energy from the hotter object.

An important property of temperature is that when two objects of different temperature are placed in contact, energy will move by itself from the warmer to the cooler, and never the other way around. In this respect the analogy between temperature (as an indicator of energy) and lake level (as an indicator of the amount of water in it) is again useful and valid: water would flow from the lake with the higher level to the lower one, too. (Notice that it doesn't matter how much water is in the two lakes, or how deep they are, or what shape, or anything else!) Similarly, the direction of energy transfer doesn't depend on amount of energy contained or the material -- just the temperatures involved.

Now consider a set of objects left together (so that they can exchange energy). The objects with higher temperature give energy to the objects with lower temperature. Generally this implies the temperatures change (we will meet an exception to this rule later), to converge on some intermediate temperature. We then say the objects are at the same temperature and are in thermal equilibrium.

In the second activity, we saw that the cups of hot and cold water were exchanging energy with the air in the room, and eventually everything will be at the same temperature -- the temperature of the room, whatever that may be.

In the first activity, we looked at the result of leaving objects undisturbed for a long time. At the end, they are in thermal equilibration at the same temperature, room temperature. Energy has moved from the warmer objects to the cooler objects, until there no longer is a difference. (We will find out later why we think woolly socks are warm and water is cold -- an important part of the story is that our bodies maintain a steady temperature well above the temperature in the room).

What a thermometer really measures is its own temperature; when we measure the temperature of something else, we place the thermometer in contact with it and wait for equilibration to occur. If objects in contact could stay at different temperatures (so that stone would always be colder than fur), a thermometer might decide not to become the temperature of the object it is touching, and we would have no way to measure or even define the temperature of anything.