Please keep in mind that part of this is my opinion, based on my experience and what I know of other's experiences then.
The minimum you need is a tank (I recommend 5 pound or larger), a regulator, bubble counter, needle valve and either a reactor or diffuser.
Either the Milwaukee MA957 or the JBJ will provide the reg., bubble counter and needle valve, and in reality the only real difference between the two models are the features and price. The Milwaukee doesn't have a check valve but has adjustable line pressure, the JBJ has a fixed line pressure but has a check valve. The need for a check valve is the topic of some discussion, as is fixed vs. adjustable line pressure, a check valve can be added to the Milwaukee for a very reasonable price and whether or not you want the line pressure to be adjustable is a matter of personal opinion more than anything.
The regulator hooks to the tank, the CO2 goes into the aquarium through the diffuser or reactor, either take care of disolving the CO2 into the water. Diffusers are more or less passive, simply bubbling CO2 into the tank, ideally in such a way as to maximze contact time with the water. Reactors are generally active, they release CO2 bubbles into a moving stream of water, usually through some kind of media that creates turbulence as well. As a rule they are more efficient than diffusers, but usually more expensive and a tad bit more difficult to set up.
I'm on my lunch so I have to quit typing, maybe someone else can take off from here, and of course correct anything I might be in error on.