There are two commonly used methods of making stained glass:
lead came, and
copper foil. Both involve similar first steps in designing the layout to allow space for joining and in cutting the glass; the differences are felt more in assembly.
The copper foil method involves using copper foil (think aluminum foil, only with copper) with adhesive on one side. You stick the foil to the edges of the individual pieces of glass, and then solder the pieces together on both sides. The solder adheres to the copper, and when it cools, holds the pieces together.
Copper foil is ideally suited for making three dimensional pieces such as boxes and lamps - indeed, Tiffany, famous for intricate stained glass lamps, pioneered the method. It is also suited for small delicate pieces, as it allows thinner lines than lead came does, but this thinness makes it unsuited for large pieces such as windows, as the thin layers of solder used in copper foil tend to sag with the weight of the glass over time in larger pieces. Copper foil is also suitable if you only have a big honking soldering iron, as the copper will not warp under heat like lead cames do. Copper foil makes working with curved and irregular pieces easy, as it is very flexible.
The lead came method involves fitting the glass pieces in to lead channels, called cames, and then soldering the cames together where they meet. Lead is malleable enough that you can shape the cames with your hands (USE GLOVES! Lead poisonning is deadly.), but the thick cames make it a sturdier choice suitable for windows and large pieces. Be careful to control the heat of your soldering iron - lead cames melt easily. Lead came is easiest to work with for straight pieces - curves and other more intricate designs require very careful and precise assembly.
The solder/lead on the finished pieces (for either method) may be treated with a wash that colours it to a brass colour or to a flat black. (Left alone, it is silver in colour - attractive, but too flashy for some pieces.)