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.