What is Hydrogen Peroxide?

Strictly speaking, Hydrogen Peroxide is an aqueous solution, H
2O
2. Fancy chemistry aside, it's an oxidizer. Typically, it's marketed as a way to "add oxygen" to your water. However, it also has another important use. It's like a nuclear weapon when it comes to bacteria.
The Hydrogen Peroxide used in hydroponics isn't the variety that you'll find at your local pharmacy. The Hydrogen Peroxide that you'll usually find in a pharmacy is typically quite dilute (about 3% is the usual strength). It's pretty safe to handle, and doesn't usually cause much concern. The Hydrogen Peroxide used in hydroponics is a whole lot stronger, usually about 35%. Be careful when you handle it... seriously. Hydrogen Peroxide at 35% will give you a chemical burn. Do not handle it without gloves. You have been warned.
Why Does It Matter?
Hydrogen Peroxide is wonderfully useful for your plants. If you have a bacterial problem, it offers you a pretty non-destructive way to wipe out all the bacteria in your system. Most importantly, it won't harm your plants. If you've ever had a nasty bacteria set up shop in your reservoir, Hydrogen Peroxide is your best and easiest bet. Since it will naturally "break down" into oxygen and water, you don't need to "flush" after using it.
Typically, if you wish to perform a bacterial reset, I would recommend a dosage of 1/3 tsp per gallon. This isn't a lot, it translates to about 18 gallons of water per 1oz of 35% Hydrogen Peroxide. Hydrogen Peroxide is pretty inexpensive, when compared to other hydroponic additives.
When Should I Use It?
Typically, I'll run a Hydrogen Peroxide cycle about every 8 weeks, just to keep things clean and controllable. Remember, when you use it, you'll be killing off all the bacteria in your system (good and evil). Many nutrients and supplements contain beneficial bacteria. Know that you'll be killing them off. For example, it would be a terrible waste to use Vegan Compost Tea at the same time as Hydrogen Peroxide. You would kill off all the beneficial biology in the Compost Tea. Typically, I'll use Hydrogen Peroxide during a water change cycle. I'll drain the reservoir, then fill with clear water. I'll add the Hydrogen Peroxide, then start up the pumps. I will usually let the system run for three days without any nutrients, as this will let the Hydrogen Peroxide fully break down. After three days, you're guaranteed safe to add your regular nutrients back into the mix.