I understand what you mean on that part. He is telling you how to convert from the AC to DC. Normally, to get DC, you do this:
AC --> Transformer --> rectifier (can be built with 4 diodes) --> capacitor (for smoothing the waves).
This gives an unregulated DC supply. If the transformer is say 12v, then the output would be a bit under 12v (change due to diodes). However, since it is not regulated, if the AC voltage changes a bit, the output will also change.
To fix this, you can do like above and then add a voltage regulator at the end (5v, 12v, etc.) The input voltage from the transformer has to be higher than the regulated voltage. For instance, if you wanted 12v out, then you might need 16v at the transformer.
A cheap/easy way to deal with this part of the circuit is with a computer power supply (from a desktop). These are cheap and easy to find used. They have regulated 5v, 12v, and -12v output and are able to provide a good amount of current. This is what I would use as a starting point for your circuit.
Maybe you can explain the experiment better or perhaps you can give we what the current (not current density) is. Also, how accurate do the values need to be? What I mean is this: say you want 2.5v, does it have to be 2.500v (almost exact) or can it be around 2.5v (say between 2.45v and 2.55v)?
How precise things need to be make a big difference on how the circuit is made. I assume that you don't need things that exact. When you run the experiment, you can use a digital volt/amp meter to see what the values are actually getting.
Controlling the volts and current make things a little more complicated, but it can be done. If you only needed to control voltage, all you would need is the computer power supply, a fixed resistor, and an adjustable resistor. It is about the easiest circuit that can be made (called a voltage divider). It won't work for you since you need to adjust the current too.
Anyway, give me as much information as you can about what you need and I'll see what I can come up with.