What is temperature?
The kinetic theory of gases defines temperature as proportional to molecular kinetic energy, and naturally encompases the idea of absolute zero; a minimum possible temperature at which all molecules are stationary.
links:
Kinetic gas theory: a brief intro
Kinetic theory derivation
Kinetic theory derivation2
Kinetic theory derivation3

A more fundemental (albeit conceptually challenging) definition of temperature is
1/T=dS/dE
where T is temperature, S entropy and E internal energy (and dS is the small entropy change resulting a small internal energy change dE, and dS/dE is the ratio between them).
This definition encapsulates the 'statistical physics' explanation of why heat flows from regions of high temperature to low temperature; because doing so reaches physical states that are dramatically more likely - or in other words - because it maximizes entropy.
To 'explain' it in a single sentence: If we take two objects at different temperatures, the hotter one will have a lower dS/dE (=1/T) and so the total entropy of them together will be increased by a flow of heat from the hotter to the cooler.
This definition works across diverse physical systems, and also gives rise to the strange concepts of infinite temperature and negative temperature.
links:
Temperature, entropy and negative temperature
Lasers and negative temperature
Is negative temperature possible?