advantages are:
The media is dissolved very slowly, so one it is running then it should go pretty much maintainance free for several months
It should keep calcium, alkalinity and magnesium pretty stable.
Compared to some methods, the actual running costs are quite low once set up.
Disadvantages:
can be tricky to get set up properly
initial outlay is quite high. Price depends how big the tank is and how sophisticated oyu wanna get.
PH swings, quite a few people have problems with mainataining a stable PH.
The basics are quite simple, reactor is full of meidia, water is slowly pumped through it and C02 is injected. The c02 causes a lower PH inside the reactor which makes the media dissolve. So long as the PH of the water coming out the reactor is lower than the PH of the tank, then its running ok and just needs to be fine tuned.
You only really need one for hard corals that use these elements up quickly.