Range of Electromagnetic Waves/Radio waves?
What basically determines that how far the electromagnetic waves will
  travel away from the source ?


EM waves will travel to the edge of the universe.


  In our world where there is air and other materials how
  electromagnetic waves would generally response to them ?


An EM wave will generally pass through air unhindered but some frequencies (such as 60 GHz) will be hindered by atmospheric absorption: -




  What will determine their range in our world.


Radio waves spread-out as they leave the source so the power thins out (just the same way as the perceived light from a distant lightbulb gets weaker). In order to be useful, the power received has to be greater than the background of thermal noise. A radio wave also has to be stronger (most of the time) than man-made interference. Lower frequencies are better because an antenna is bigger and naturally picks up a bigger signal (more power). The disadvantage is that longer wavelengths need much bigger antennas than smaller wavelengths. 


More transmit power = greater distance. 
Antenna directionality = more distance.
Lower frequencies = bigger RX antenna = more RX power (or more distance).
Some frequencies are better at penetration through objects than others.