Screen printing aka silk screening is the technical term for printing on t-shirts, sweatshirts, etc. Screen printing designs are vector based, not pixel based.
A vector based image is made up of points, lines and curves filled with color. A pixel based image is made up of tiny squares of color.
Most images you find on the Internet are pixel based. So converting that to a vector based image takes a lot of work.
The Printing
Here are some pictures for some shirts I made a few years ago for my Dad's slow-pitch team. This is a 1 color, 1-sided print job.
Step 1 - Create your design on the computer and print it out on film or vellum. Then tape your image face down onto a screen and burn it. Wash out the image area with the power-washer. (sorry, didn't take a picture of that part)
Step 4 - Lift the screen and carefully remove the shirt from the table.
Step 2 - Cover all non-image areas of your screen and flatten your shirt on the turn-table.
Step 3 - Coat your screen with your desired color of ink and then push it through the image area with a squeegee.
Step 5 - Place you shirt on the belt of the dryer.
Step 6 - Remove your shirt, tah dah! (the temperature is what cures the ink)
Step 6 - Remove your shirt, tah dah! (the temperature is what cures the ink)
The Cost
I can't tell you what it will cost to get your shirts printed, because that all depends on the printer's specifications. But I can give you a little background if you're interested.
Screen printing cost is based on several variables:
(1) Shirt type (this price typically stays the same regardless of quantity)
(2) Shirt quantity (more t-shirts = lower cost)
(3) How many colors your logo contains (more colors = higher cost. Each color needs a separate screen, see photo to the right.)
(4) How many areas you print on (more areas = higher cost)

