Understanding the configuration of channels in the 10-20 EEG montage
EEG 10-20 system. source: Wikipedia


  
  How many actual channels are there?
  


21 in the figure, i.e., the number of active electrodes.


  
  What's the difference between reference and active channels?
  


The active electrode is the electrode under investigation. Basically this is arbitrary. Consider electrode F2 and F3. When you measure the potential difference between them and F2 is the active electrode and F3 the reference, the signal will be exactly identical as in the reverse situation, only the polarity (the sign of the voltage) is reversed. That's all there is to it. The active electrode is typically the electrode linked to the location you wish to record. The reference is elsewhere.


  
  What is the difference between a bipolar and a normal
  (unipolar/monopolar) setup?
  


In a regular unipolar setup, each of the electrodes is either measured against a distant reference (e.g. the earlobe) or against the aggregate of all the electrodes. Either way, the reference is distant. This means that the signal will be high, but artifacts will be large too, because artifacts from both the electrodes will be merged into the signal. Consider a reference in the neck; neck musculature activity will add artifacts (EMG) to the signal. Likewise, a reference in the chest area will add ECG to your signal.

In a bipolar setup adjacent electrodes are recorded. This means that large artifacts occurring through eye blinks or whatsoever are recorded by both electrodes and as their polarity is opposite, the large artifact responses will be cancelled and the signal is much cleaner. However, the downside of the coin is that commonalities in the signal, which will be larger when the electrodes are closer will also be subtracted and hence your signal will decrease accordingly. 

I have used bipolar setups when measuring eERGs and eCAPs, i.e., electrically evoked activity. The electrical stimuli generated large artifacts and actually so large that their amplitudes were approximately similar between active and reference electrodes. The reduction in artifact by far outweighed the loss in signal. It depends on the situation you are at.

An authoritative work in this field is listed below.

Reference
- Pivik et al., Psychophysiol (1993); 30: 547-58