Screen printing on apparel allows lots of different possibilities as far as artwork goes, but for the best results we typically recommend spot colors. Spot colors are just solid colors, with no shading within that color. Shading can happen with halftones (a series of tiny dots) but you need high resolution artwork to get crisp halftones, and the cost of printing climbs with more colors, and more artwork prep on complex designs.
Vector art is always the safest bet for printing. Vector art maintains the ability for the art to be scaled infinitely without loss of quality. Pixel based art will blur and develop rough edges when blown up to much. Vector formats are usually created in Adobe Illustrator or Corel Draw, and are saved as .ai, .eps, .svg or .pdf formats. Vector art allows the printer to scale the art to the right size for your job, and pull the separate colors out easily for outputting to film.
Pixel based artwork (.psd, .jpg, .tif, .bmp, etc.) can work great to as long as the resolution is high enough. Anything above 200 dpi, on a canvas sized to the approximate scale you'd like to see it print will work great. The main situation we need to avoid is blowing up tiny graphics from the web to a printable resolution, this can cause rough and distorted edges that will make your print look low quality.
If you're not sure what you have, the best bet is to send it on over to us and we can quickly let you know if we're good to go with your file, or if we'll need to do any extra work to get it ready for print. There is always a way to get your idea on a shirt!