Your skin is a terrible thermometer
On a cold morning a metal railing and a wooden bench sit outdoors at exactly the same temperature, yet the metal feels far colder to your hand. Your skin does not sense temperature directly — it senses how fast heat is leaving it. Metal carries heat away from your fingers quickly, wood slowly, so the metal feels colder even though it is not. Before we can do any physics of heat, we need a measure of hot and cold that does not depend on which finger, or which day, is doing the judging.
Temperature and thermal equilibrium
Put a hot object and a cold object in contact and leave them. Energy flows from the hotter one to the colder one until, after a while, nothing more changes: they have reached the same temperature. That final unchanging state is thermal equilibrium. Temperature is the single number that decides which way heat will flow — from higher temperature to lower, always, never spontaneously the other way.
Notice the subtlety: heat only flows while there is a temperature difference. At equilibrium the two bodies are still full of jiggling molecules and still exchanging energy back and forth — but the two flows are equal, so there is no net transfer. Equilibrium is a busy standstill, not a dead one.
The zeroth law — why thermometers work at all
Here is a fact so basic that physicists nearly forgot to state it, then named it the zeroth law of thermodynamics: if object A is in thermal equilibrium with C, and B is also in thermal equilibrium with C, then A and B are in equilibrium with each other. In plain words, 'has the same temperature as' behaves like ordinary equality — it can be passed along.
Temperature scales and absolute zero
The Celsius scale pins 0 °C to water's freezing point and 100 °C to its boiling point at sea level, then divides the gap into a hundred equal steps. Handy for weather, but it starts at an arbitrary place. Physics prefers the Kelvin scale, whose zero sits at the coldest temperature that can possibly exist. A one-kelvin step is exactly the same size as a one-Celsius-degree step, so converting is just a shift.
Kelvin from Celsius: add 273.15. So 0 °C = 273.15 K and 100 °C = 373.15 K.
The zero of the Kelvin scale, 0 K = −273.15 °C, is absolute zero: the temperature at which molecular motion is as small as it can be. You cannot go below it, and there are no negative kelvins. (Honesty check: motion does not entirely stop at absolute zero — a residual quantum 'zero-point' jiggle remains — but that is a story for a later quantum track.)
What temperature measures underneath
Zoom in and matter is a swarm of jiggling molecules. Temperature turns out to be a measure of the average kinetic energy of that random molecular motion. Hotter means the molecules, on average, move faster. For an ideal gas this link is exact and beautifully simple.
For an ideal gas, the average translational kinetic energy per molecule is proportional to the absolute temperature; k_B is the Boltzmann constant. We derive this fully in the Kinetic Theory track.