After masking off the parts of the copper-clad board you want to keep, you need to remove the parts you don't want to keep. This is usually done by chemically etching away the copper. There are a lot of different chemical techniques for doing this, each with its own advantages and drawbacks.
None of these chemicals is incredibly dangerous, but they can all be toxic or caustic, and should be treated with care. Eye protection and gloves are a very good idea. Before you start, make sure you know how dangerous each chemical is, and figure out what you will need to do if you spill it or get it on yourself. Washing with plenty of water is usually a good start. For some chemicals you may want to keep a neutralizing agent handy. An MSDS (Materials Safety Data Sheet) for the chemical will give you some basic information.
This is the most common hobbyist etchant. Ferric chloride, FeCl3, is a brownish substance. It's usually sold in a bottle (dissolved in water, perhaps with a little acid or peroxide) or as a powder (which you have to dissolve in water).
When in solution, ferric chloride is a ferric ion (Fe3 ) and a chloride ion (Cl-). The ferric ion reacts with the metallic copper on the circuit board in a redox reaction, producing a ferrous ion (Fe2 ) and cuprous or cupric (Cu1 or Cu2 ) copper. The chlorine is just along for the ride. The copper ion, unlike the metallic copper, is soluble, so it leaves the circuit board and goes into solution. The reaction products form a black sludge which settles to the bottom of the etching tank. After etching enough copper, all your Fe3 is used up and your solution is full of Cu1 , and you need to get more etchant.
HydroChloric Acid / Hydrogen Peroxide
Mixing about 1 part HCl (Which can be found at most hardware stores, also known as Muriatic Acid. Ask for concrete cleaner.) into 2 parts Hydrogen peroxide (normally used for cleaning cuts) you can make a fairly powerful etchant. Use gloves and don't breathe the fumes though. This will etch a 3"x5" board in less than 10 minutes. No need to heat it up. I usually like to drill a small hole through the board on a corner and thread a wire or nylon string through to help agitate / remove the board.
When the board is done etching, the etchant will probably look like green kool-aid, from the copper content in it. This stuff is highly corrosive and will burn skin, which is why you should wear gloves. But it is easy to handle, and fairly easy to dispose of. the etchant is easily deactivated with baking soda. Pour enough baking soda into it slowly (to keep it from boiling and overflowing... remember what happens with baking soda/vinegar? ya...) until it is a solid mass, then leave it in the sun to dry. You should contact your local authorities to find out what you should do with it next. Whatever you do, DO NOT dump the stuff down the drain, it will eat through your pipes just like any of the other etchants.