I have built this reactor. There is a lot of explanation in the thread that CaPhil linked to.
The reactor needs to be mounted vertically, with the end with the tubing inserted (right hand side in the pictures) up. The top barbel is connected to the outlet port of your canister filter, and the bottom barbel goes to the tank.
The longer piece of airline tubing is connected to your CO2 system. The tube runs most of the way to the bottom of the reactor. The CO2 bubbles come out near the bottom, and try to travel up to the top of the reactor. Since water is pumped into the top, and flows down, the CO2 bubbles are rising fighting the current, so it takes them longer to reach the top, if at all. My reactor is not transparent, so I cannot see exactly how the bubbles are acting, however I rarely notice any buildup of gas at the top of the chamber.
The shorter piece of airline tubing installed in the top is to allow you to off-gas any built up gas that has failed to dissolve. You will know that you are getting a pocket of gas forming if your reactor starts making noise, as you will hear the water dropping down through the gas pocket to hit the water level in the reactor. Tomm Barr refers to this gas as "recalcitrant gas". You will want to attach a valve to the off-gassing tube. You can either manually bleed off gas that has formed by opening the valve, or you can then connect the outlet side of the valve to your filtration system somewhere upstream. You will want to introduce the off-gassing tube into the system at 90 degrees to forma venturi that sucks the built up gas out and injects it back into the system.
I have not personally connected the off-gassing tube to a venturi, as I do not notice any gas build up in my system. The only time I need to operate the off-gas valve is when I clean out the reactor and am refilling it. When you first fill the reactor a very large volume of air is trapped, and it takes forever to clear without the off-gassing valve.
I have my reactor hooked up inline with my canister filter. Tom Barr is showing it being driven by a submersible pump (which would be in the sump, if you have one). I do not know that a power head pulling water through would work. The problem would be that you would be creating an area of low pressure in the reactor. I would think that the result would be to encourage the buildup of "recalcitrant gas" in the reactor, as the low pressure would encourage all the dissolved gasses in the water (CO2, O2, N2 etc) to come out of solution. I would be leary of trying this with a power head myself, and would definitely want to have the off-gassing tube hooked up to a venturi.