Screens are made by putting a chemical emulsion on a mesh surface. Vector artwork (in most cases is the best file to use) is taken from Adobe Illustrator or Freehand and printed out on a film type paper or vellum. The screen is then exposed on a light table with the artwork under it. The light solidifies the chemicals around the design, and where the light failed to pass through, the chemical breaks down. The screen is then rinsed out and what is left is the area where the light hit.
The shirt is placed on a pallet and each screen swings around over the top of the shirt. The screen is then brought down and placed on top of the shirt. A squeegee is then pulled over that screen's ink colour which pushes the ink onto the shirt fabric. That screen is then lifted, carefully off of the shirt (If the shirt moves or is stretched, the next colour will be out of registration). The pallet with the shirt is then moved under a flash unit where it dries. Upon curing, the shirt pallet is brought back and the next color screen is swung over the top of it.
When screen printing, one color at a time is applied and therefore the cost will increase with each additional color. Likewise a separate screen is required for each color.
Screen printed garments wash well, and should not be confused with heat transfer printing which does not look as good or last so well. Many t-shirt printing shops use this cheaper method rather than screen printing.
This is a picture of what t-shirts are printed on!