Regrettably, There is just not enough space here to go into any major detail on the chemistry, and this is covered in far greater detail in the those two links so I will breeze over most of it. Suffice it to say that the articles referenced at the bottom of this step should be your primary resources.
Simply put, given the right amounts of oxygen, HCl (Hydrochloric acid), copper, and water, we will have a perfect amount of etchant.
The reaction we want to happen is:
Cu + 2 HCl + H2O2 --> CuCl2 + 2H2O
The copper comes, initially, from the circuit board itself. Hydrochloric acid is commonly available as a pool cleaner, sold as Muriatic Acid. Mine can as 31.45% (20 degree Baume, or 10.01 Molar). Hydrogen Peroxide can be found in higher concentrations at health food stores. You want 35% Food grade (11.6 Molar); the 3% kind you get in the first aid section just won't cut it. You also need a jug of distilled water, as tap water is just too impure.
For my 2 L tank, I only need 1.6 Liters of etchant. The ratios in the above formula, and assuming my etchant should be 3 Molar HCl at its final volume, I came up with .48L HCl, .92L H2O, and .20L H2O2 as my reagents. Add the water, then the Peroxide, then the HCl. Always add acid to water, and never the reverse (As the mnemonic goes, "Do as you oughta, add acid to watah [water]"). And do it outside, in a pyrex dish. It will produce heat and fume fairly violently at first, but as you begin to use the solution to etch boards, the fuming will decrease as it turns to Cupric Chloride.
Note: Do not make the mistake of mixing your etchant in your homemade etch tank. Absolutely do not etch too many boards (side by side) or mix raw copper into your etchant, as i did, especially in an enclosed sealed tank. The reaction will be violently exothermic and fume viscously. It will melt, crack, or break open your etch tank from pressure and heat that is far beyond what is normally produced with a controlled stable etchant. With a sufficient air bubbler, you won't need to mix in the copper initially, anyway. The solution will effectively make itself.
After mixing the etchant, it is time to etch! If you have built your etchtank carefully, all you need to do is slide the pcb into the clip on the lid and place it in the tank. The etch progress can be observed through the window in the side. The hanger I currently use needs some improvement as the boards need to be turned and replaced off halfway through to etch the part covered up by the clip.
The boards will etch rapidly at first, in under a minute, and as the solution begins to become saturated, the etch times will increase to five minutes or more. Once it becomes unreasonably long, you simply need to add the right amount of acid and/ or run the bubbler for a period to regenerate the solution and decrease etch times. Again, consult the above paper for how to do this properly; it is a very good reference.
When you can see the board is fully etched, remove the lid and pull out the board with plastic tongs. Let the excess etchant drip into the tank, and place the board in a plastic tub to rinse it off. Then it's on to the Scotch Brite pad and some acetone to take off the layer of toner, and the copper traces should polish up to a bright shine.
When the tank is not in use, the etchant should be stored in an airtight glass vessel that will not be used for food, out of the light, and properly labeled. Under no circumstances should the etchant be poured down the drain, as it is both toxic (the high copper content) and corrosive to metals. If you do happen to spill any, or need to dispose of it, the solution can be neutralized with lime, and mixed with concrete for disposal. Be sure to dispose of it properly, according to your local hazardous waste policy! Where I live they have a facility will take in waste like this for free.
It is O.K. if some of the rinse water ends up in the sink, assuming you do not have copper pipes and that you leave the water running afterwards long enough to flush it all out of the system. I cover and resuse even my rinse water, treating everything that comes in contact with the acid as hazardous waste. So don't go pouring anything down the sink, seriously.