The bulb is the part that actually produces the light.
The ballast is the electronics that converts normal power from your plug to the high-voltage the bulb needs to operate. Typically built into a little black box, with wires running from it to the light, and it's mounted somewhere in your hood.
Many aquarists find they need more light, but don't have room for additional standard bulbs in their hood, don't want to pay for more expensive high-output bulbs, or find it difficult to mount a different size or kind of bulb. That's where ODNO offers an alternative.
ODNO is taking a ballast designed for *more than one bulb*, and wiring two or more of the outputs together to drive a *single bulb*. This feeds more power to the bulb, and increases the brightness. The advantage is that you get more light from the bulbs you already have, just by changing the ballasts.
As for my idea, that involves recycling the ballast from spiral lights, like these
. These replace standard light bulbs, and have a fluorescent bulb and a tiny ballast built into one unit. When the light burns out, you normally throw away the whole thing. Or if you're me, you crack it open, take out the ballast, and see what you can do with it.
The link I showed you demonstrates how you can take a ballast from a 23W spiral light, hook it to a 15W bulb, and get 23W of light from it. The only catch - since you've taken the ballast out of its original protective casing, you must mount or re-case it in such a way that will not come in contact with water or be a shock hazard.
I made no assumptions about your level of comfort with electrical work when I replied, I only listed some other options. You should ultimately go with what you're comfortable with.