Basics of battery charging and discharging
Does "discharged" mean 0 volts and so I can't draw anymore current
  from it ?


You need to read the data sheet for the battery i.e. something like this should pop-up: -



This is for a 2500 mAh battery of terminal voltage 1.2 volts and the data sheet tells you that the "end-voltage" is 1 volt. In other words from full-capacity to end-capacity the voltage has fallen from 1.2 volts to 1 volt. It also states that this assumes a constant 500 mA discharge.

So, you need to read and understand the data sheet.


  Can I recharge this battery by simply applying a constant voltage of
  1.2 V across it terminals ? Or it should be charged by some constant current ?


You need to read the data sheet and take advice from it.


  Is the battery considered to be fully charged when its voltage reaches
  1.2 V ?


Normally, yes but under a prescribed load.


  What differentiates a non-ideal battery from ideal battery in these
  cases ?


The voltage into an open circuit does not really tell you much about the state of charge but, as a battery discharges, the internal series resistance increases hence the terminal voltage reduces with time when under loading conditions.