Thermal energy
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. Because there is motion, there is kinetic energy; there also is elastic energy, because the motion gives rise to 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 thermal energy, whose presence is detected by measuring the 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 thermal energy that was added to 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 -- in this case, 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. No matter how they started, at the end they are in thermal equilibration at the same temperature -- again at "room temperature." Energy has moved from the higher temperature objects to the objects at lower temperature, 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).
Thermal Equilibrium
When we talk about thermal equilibrium of a system, we are assuming that the system is isolated from "the rest of the world," so that the system does not exchange energy with the rest of the world. Then energy moves around within the system until thermal equilibrium is reached. The inside of your refrigerator is an example of a somewhat isolated system. It maintains a constantly cold temperature (we're ignoring how that is maintained). If you put a container of hot food into that system, energy moves from the food to all the cool things around it (the rack, the milk, the refrigerator walls ... ) lowering the temperature of the food and very slightly raising the temperature of the other things, until they are all (food included) at the same temperature. In most cases in these activities, you will treat your classroom as the isolated system and students will look at temperatures of different things coming to thermal equilibrium within that system.
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. It is important that equilibration takes place, and that
the result is objects at the same temperature: if
objects in contact could stay at different temperatures
(so that a stone would always be colder than a fur coat),
a thermometer might also never come the temperature of
the object it is touching, and we would have no way to measure or even
define the temperature of anything.