On the particle level, how is heat transfered between particles?

Heat is transferred through conduction (convection is really just a form of conduction). When 2 particles come in contact with each other they will use thermodynamic principles (the zeroth law I believe)  in order to obtain a mean of their temperatures. When any 2 objects come in contact they will transfer energy in some form between each other. Atoms vibrate dependent on their energy (heat) so there will be more contact to their movement, much like one of the answered gave a reference to the elastic collisions.

Particles vibrate and that vibration rate increases with temperature.  A particle at absolute zero does not vibrate at all and .  The more a particle vibrates, the more frictional heat it generates and transfers the heat to other particles via conduction.  

The second law of thermodynamics states that heat is transferred from a hot body to a cold body.  A colder object can NEVER give up its heat to a warmer object.

Simply stated then, heat is transferred between hot particles and cold particles due to the second law of thermodynamics.