Hey, look, I'm new but a topic I know a lot about! I'm an ex ignition systems engineer, and I used to work as a test engineer at NGK here in the US. CDI is capacitive discharge ignition.
This is a VERY complicated topic, but I seem to be pretty good at analogies. Quick background: an ignition coil is an inductive device, you apply voltage to one side of the coil, and you generate a magnetic field, when that field is collapsed very quickly, the secondary side of the coil will generate voltage that is proportional to the winding ratio on the primary and secondary (12volts in, 40,000volts out). Capacitance is created whenever you have two metal plates within close proximity of one another. For example, an ignition coil in the head (like in a coil on plug engine) creates capacitance because of the close proximity between the coil and the head. Capacitance can be added via special boxes or even some spark plugs (pulstar, if they are still around, but don't use them...) So, what does that mean for spark? Well, capacative discharge occurs very quickly over a very small amount of time (think nanoseconds), while inductive discharge occurs over a longer time, think microseconds.
So, analogy time. Think of a coil as a gear reduction with a flywheel on it. The flywheel has a clutch that engages and disengages a water pump. When you apply a 12 RPM input to the gear reduction, you get 40,000RPM on the output, but it takes a while to spin up. So, when you start spinning the input and the flywheel is spinning, this is like the dwell time. Once the flywheel is spun up sufficiently, the clutch engages the water pump, and water starts to flow into a hose. At the end of the hose is a valve that will open at a certain pressure (this pressure changes, and is analogous to the required voltage to breakdown the spark gap). "There is some stretch in the hose, so when the valve opens there is a little bit more flow for a short time, once that's done, the pump keeps pumping until the flywheel runs down." The small stretch in the hose is like the capacitance of a standard ignition system. Once the valve is opened, the flywheel energy pumps out what it can, and then it's done. A CDI system is like adding a big pressure tank to the end of the hose. Once the pressure starts to build, more and more water is stored in the tank. Once the valve opens, a HUGE tremendous amount of water comes out, but for a very short time. Once the tank is done, the flywheel/pump still pumps a little bit out. So, the pressure tank is like capacitive discharge, and the flywheel pump is inductive discharge.
So, what's the advantage? Well, the problem with a spark plug is something called quenching. When spark occurs, there is a very small flame kernel that is susceptible to quenching via nearby relatively cool metal, like the plug, the head, etc. Increasing ignitability reduces this quenching effect, which is actually the benefit of fine wire spark plugs. CDI helps this by providing a lot more energy in a very short time. Some CDI systems help even more by allowing more current on the primary side (hence the need for bigger wires).
So anyway, that's what I can add, hopefully it helps.