Now clearly the collection of atoms in a gas can perform many different types of motion. We could imagine all the atoms standing completely still (or as still as they are allowed by quantum mechanics) or we could imagine half of them at rest and the other half moving back and forth in oscillatory motion. These are not the types of states of matter which we describe through thermodynamics. We wish to describe the state which ensues in a system of particles after a well defined set of external conditions have been in effect for sufficiently long time that these average properties have converged to their final values. We call this state the state of thermal equilibrium.
An example of a thermodynamic system which is not in thermal equilibrium is a gas which occupies one half of the volume V which is available to it. Clearly there will be a macroscopic evolution of the volume of gas to a state where it occupies all of V at which point the gas may be in its thermodynamic equilibrium state.
Another example of a system which is not in thermal equilibrium is a cup of hot coffee on the breakfast table. If you left it there for several hours it would cool off until it reached thermodynamic equilibrium with the surroundings. If there were two cups of coffee on the table and they were both left to evolve into thermal equilibrium with the same surroundings then clearly the two cups of coffee would be in thermal equilibrium with one another as well . This postulate is sometimes called the zeroth law of thermodynamics.
From the zeroth law of thermodynamics comes the idea of a thermometer and the concept of temperature. We can classify the thermodynamic equilibria of even the most complicated systems in terms of the state of a simple system with which it is in thermal equilibrium. Thus if we can classify the thermodynamic state of a simple physical system be a number then we can use this same number to classify the thermodynamic equilibrium state of even the most complicated of systems.
Naturally we have lots of freedom for specifying a temperature scale. We could for example choose the temperature to be the length of a bar of a specific material because as we shall see the dimensions of solids vary with the thermal equilibrium state of the solid. This so called thermal expansion effect is however a complicated which depends on details in the atomic interaction potential which we do not want to build into our definition of temperature.
Instead we choose to define the temperature in terms of the simplest possible system which has a thermal equilibrium. Specifically the Kelvin temperature scale used in physics is defined
as being proportional to the pressure of a very dilute gas held at constant volume. This implies that zero temperature is the
temperature of a dilute gas which
is so cold that it exerts a vanishing pressure on the walls of
the volume that contains it. We need a second fixed point
to fully define the temperature scale and for this physicists have chosen the so-called triple point of water. The triple point of water
is where solid, liquid, and gas of 
coexist. It is a common
and well defined thermal equilibrium state which we label by the temperature 273.16 K. The Kelvin temperature scale is thus defined as
where 
is the pressure in the fixed volume of gas when it is
in thermal equilibrium with a mixture of water ice and water vapor
at its triple point. It is an advantage of this definition that it is
independent of the detailed properties of any one material.
Other temperature scales in use for historical reasons
are defined in terms of the Kelvin ideal gas temperature scale.
Specifically the Celsius and Fahrenheit temperature
scales are defined as
