Short answer: What is measured by a thermometer.
Defining something by providing a procedure for measuring it is called an operational definition. To complete the definition of temperature requires a detailed description of how to build a standard thermometer and how to calibrate the instrument. Calibration involves subjecting the thermometer to known conditions and insuring that it reads the standard or stated value.
The first attempt at building a thermometer dates back to Galileo (around 1600). Over time thermometers improved in their ability to agree with one another and to get the same reading when exposed to the same conditions. Making accurate thermometers is very important for many applications today. The International Temperature Scale (ITS-90) was developed to assure accurate temperature measurements. Published in 1990, the ITS replaced its predecessor, the International Practical Temperature Scale. (1)
Many of the formulas used by scientists and engineers contain the temperature. There is a theoretical understanding that relates what we measure with a thermometer to what is going on inside the material whose temperature we are measuring. This theoretical temperature is often called the thermodynamic temperature (more on this is a moment). Is is the role of the ITS to make sure that what the thermometer measures accurately reflects the thermodynamic temperature . The difference between the ITS-90 defined temperature and the thermodynamic temperature is less than about 0.01% when expressed in Kelvin, and at room temperature better than 0.0025%. (2) The ITS-90 serves as a bridge between what we measure and our theoretical understanding of how temperature relates to what is happening within the material whose temperature we are taking.
Temperature Scales
If you are in the United States you measure temperature in Fahrenheit. The easiest way of thinking about this scale is that a Fahrenheit thermometer is calibrated so that water freezes at 32 degrees Fahrenheit (oF) and water boils as 212 oF, so long as the water is at a standard atmospheric pressure (roughly that at sea level). If you live in most other-places in the world you measure temperature in degrees Celsius (or Centigrade, same scale). A Celsius thermometer is calibrated so that water freezes at zero degrees Celsius (oC) and water boils at 100 oC, again both calibrations done at standard atmospheric pressure. Both of these scales measure exactly the same thing, the thermometers are simply marked differently
Both Fahrenheit and Celsius scales include negative temperatures. There is however a lowest possible temperature which occurs at -273.15 oC or -459.67 oF. To keep formulas involving temperature simple, scientists often use a temperature scale where zero is the lowest possible temperature scale. The easiest way to do this is to simply add 273.15 to the Celsius temperature reading. This defines the Kevin scale. Note that a change of 5 Kelvin (5 K no degrees in the Kelvin scale) is the same as a change of 5 oC!
Thermodynamic Temperature
Consider a balloon filled with “air.” The air consists of molecules of nitrogen (two nitrogen atoms bound together or N2), molecules of oxygen (O2), some carbon dioxide (CO2) and various other gases. These molecules don’t just fall to the bottom of the balloon, they are energetically moving around, bouncing off each other, and some are bouncing off the balloon! The typical nitrogen molecules (at room temperature) are moving faster than 1,500 feet per second or over a 1,000 miles per hour! Heat up the gas, the average molecule moves faster. Cool the gas, the average molecule moves slower.
A liquid is similar to a gas in that the molecules are free to roam and are bouncing off each other. Just as in a gas, as the temperature increases so does the speed of the average molecule.
In a solid, the atoms are fixed in place, otherwise the solid wouldn’t be solid! However the atoms can still vibrate. Since the atoms are moving (back and forth) they are moving. As the temperature increases the average molecule vibrates faster.
Molecules in a gas or a liquid can also vibrate and rotate as well as zing along (what the physicist calls translational motion). Think of N2 as two nitrogen atoms connected by a spring. The spring (not a rigid rod) allows the atoms in the molecule to vibrate. Similarly the barbel shaped N2 molecule can rotate. These rotational modes and vibratory modes only happen at certain energies. When the temperature increases more of these vibrational and rotational motions can come into play.
Temperature and Kinetic Energy
Kinetic energy is the energy associated with movement. It is dependent on the mass of the object moving, and the speed at which it is moving. Double the mass and you double the kinetic energy. If you double the speed and you get 4 times the Kinetic energy. If temperature is related to the movement of the atoms and molecules it should also be related to the kinetic energy of the atoms and molecules. It is.
In the case of a simple gas of non-interacting atoms (like Helium) near room temperature (where it is a gas) the temperature is just a measure of the average kinetic energy of the Helium atoms. The thermodynamic temperature (measured in Kelvin) is related to the average kinetic energy of the helium atoms (assuming they are free to roam in three dimension) by the formula
average kinetic energy = 3/2 k T
here k is the Boltzmann constant 1.38065×10−23 Joules/Kelvin and the average kinetic energy is measured in joules. In this case the temperature is simply a measure of the average translational kinetic energy of the Helium atoms.
Experiencing Temperature
First an exercise. Place your hand on a granite counter-top or a metal stove. Now put you hand on a pot-holder (or a piece of fabric). Which feels colder? Don’t read below, go do it and then come back…
You probably noticed that the counter-top felt colder than the pot-holder. However in reality they are probably very close to the same temperature! Unless it is really hot, the temperature of your hand is above that of the counter-top, stove and pot-holder. Heat flows from the hotter object to the colder object. The metal stove or stone counter-top “sucks” the heat out of your hand faster than the pot holder. So the counter-top (or stove) feels colder.
You feel the exchange of heat between your body and the environment, you do not directly sense temperature itself.
References
(1) https://www.nist.gov/sites/default/files/documents/pml/div685/grp01/ITS-90_metrologia.pdf accessed on 1/13/2
(2) http://www.nature.com/nphys/journal/v12/n1/box/nphys3618_BX1.html accessed on 1/13/2017