Temperature is actually a rather
subtle concept.
Currently there are two well established, consistent but independent definitions of temperature:
1. From classical thermodynamics: temperature is equal to the derivative of internal energy with respect to entropy.
2. From statistical mechanics: temperature is defined by the shape of the Maxwell-Boltzmann distribution for the system if it is a system with mass, and by the Planck distribution if it is a system without mass (i.e. a system of photons).
These two definitions are consistent, as it can be shown that for systems in which both are well defined, they are equal -- but both are not always well defined. For example, in plasmas, it is possible for the temperature of the system to be different for different directions (x,y,z)!
One rather interesting side effect of the coexistence of these two systems: if the system has an energy maximum (and a few other interesting properties), it may have a thermodynamically negative temperature.
One more note about the preceding discussion on this topic. According to the zeroth law of thermodynamics, there is an absolute temperature scale. That scale is the kelvin scale. Because the scale is not set by measuring reference points and then dividing the area between, kelvins are not measured by degrees. Temperatures given in units of kelvin are simply "kelvin", not "degrees kelvin", and there is no degree mark beside units of kelvins.