What the other answers are missing is something that needs to be made explicit -- when the system is not in equilibrium, there is not a single temperature. In other words, for a non-equilibrium system, there can be multiple temperatures.
For instance -- if a system has the translational, rotational, vibrational and electronic modes all excited but not in equilibrium, there can be 4 temperatures. One for translation, one for rotation, one for vibration, and one for electronic. As the system reaches an equilibrium (through colliding molecules), the 4 temperatures approach one another and eventually all reach the same, equilibrium value. Assuming the system reaches equilibrium.
In the example of the human body, this is also true. The body has one temperature, the thermometer has another. Two temperatures are correct for the system. Over time, the body cools (very slightly -- not enough to be noticed really) and the thermometer heats up. Eventually, the system reaches an equilibrium and you can read the temperature.