the most common is silk screen prints.
its simple, just a fine silk screen with the design imprinted on it. the screen is first coated with a layer of gel that will harden when exposed to light. the design to be printed on the shirt is printed in black on clear acetate and placed over the screen, so when exposed to light, the gel beneath will stay in its current state, and is washed away.
the screen is positioned on the shirt, and the ink is spread on with a squeegee, then scraped off. when the screen is lifted, the design is printed on the shirt.
for multi-color prints, each color gets a different screen, with a different section of the design printed on the acetate. the colors are put on in layers, being flash heated to dry the ink between each color being added.
the same method is used commercially, just automated and on a large scale.
There are several different ways to screen print shirts (the actual term. Nobody really uses silk anymore. The name just sticks). They got those cheap kits at Office Depot you can hook up to your printer. It's only 20 bucks but what you spend on ink and a compatible printer, tshirts, etc. it's not even cost effective and it's very limited on what it can do.
I REAL kit would be a 3-5 color process machine or highter (5' x 5' area or more) where the screens are on wooden frames and act kind of like stamps. Different designs per color (same shirt) and each shirt rotates in a circle to each frame. Each frame has a synthetic fiber sheet stretched tight, with a bunch of tiny holes where the paint passes through. You can squeegee it by hand or the machine can do it for you depending on what you have. It sounds like a tedious and long process because it is. You have to create and design through a special software too, then create the the physical design and film for your frames, print the shirts and makes sure everything is adjusted, and flash it, which just means passing it through the heating device to dry it out. It's a lot of work yes, but once you get into it you get hooked and obsessed with perfection. It's an art and you tend to think of yourself as a bad *** once you get it down and you see what you created on your own...lol