opamp comparator circuit question battery discharger tester arduino
why there isn't a current limiting resistor between the output of the
  op-amp and the base of the NPN? and won't this current from the op-amp
  added to the total capacity of the battery?


The emitter is forced to be at Vref (due to feedback) so, in effect the circuit can be regarded as an emitter follower and, as such a base resistor is not needed. However, given that a relay can disconnect the main power source to the TIP41C, I would think it's a good idea to use one.


  For an arduino analog pin with limited resolution.(5-0v 1024;
  5/1024=0.0048v) How accurate will it be?


You are confusing resolution with accuracy. If you want to understand the accuracy take into account INL, DNL, zero offset and gain slope errors stated inside the data sheet for the Arduino.


  A relay is used to end the discharging process. Will it be more simple
  by using a LOW digital output on the base of the NPN instead? Or maybe
  it's also okay to use a LOW digital output on the Vref of the op-amp?


I would go for clamping Vref to 0V to limit the current but bear in mind the input offset voltage of the LM324 can add a small error and cause several tens of microamps still to flow. Maybe choose a much better op-amp. I would use a MOSFET to clamp Vref or maybe an analogue switch.