Screen printing involves using a fabric with holes in it (like a window screen but finer) and some of the holes are blanked. The screen is laid down on the surface to get an image - shirt pulled tight, paper, canvas, metal, glass - and a thick paint or ink is pushed across the screen and through the holes with a rubber squeegee.
Masking the holes can be done manually or with photo sensitive emulsion.
To do multiple colors, several screens are required and some alignment method is required.
Because of this doing a one off in multi colors is about as costly as doing ten or more and they will want money up front.
Other choices are printing the image on special paper with inkjet inks and ironing the back of the paper to transfer the ink - multiple colors, hard to do intense colors
or printing directly onto the cloth with an inkjet printer designed for the purpose - you can see these (I believe) in mall kiosks that will take a picture and print it on the shirt - although some or most may do it by printing with common printer and ironing on. Go and check.
The screen printers I know realize most people don't know how to make the screens themselves, so turning the art into a usable screen is part of the service they provide. Some prefer finished camera ready art work, or a hi-res jpeg image file. If you don't have that, then they can have a staff artist render your drawing so it's suitable for screening, at a cost, needless to say. If there are colours then you assist in colour selection. For large orders they will make a prototype or sample, then you approve the sample or ask for changes, then the final screenings will be done.
You can get varying service from different screeners, so you might want to shop around, especially if it's for a band shirt or a team shirt. Ask to see samples, ask for references, check with the better business people for complaints and don't pay everything up front, pay a bit at the beginning, then pay the remainder on delivery.