Screen printing machines are pretty simple devices. The most basic screen printing machine for t-shirts holds a single screen so that it can be dropped down onto a printing surface holding a t-shirt, consistantly in the same position time after time. This allows the mass production of printed shirts.
A multi-color printing press is a little more complex. It holds multiple screens on arms that revolve around a center axis. Each screen is set up with a different color ink. The first screen is dropped onto the shirt, and a stroke is made laying the first color of ink on the shirt. That screen is lift, and the next screen is rotated into place, dropped, ink laid down for the second color. There may or may not be a drying step between screens, where the surface of the ink that was just laid down is dried using a flash dryer.

Screenprinting machines for t-shirts for sale:

AWT Econo-Tex Tabletop Printers
AWT Econo-Tex II Floor Model Printers
Blick 1-Color Garment Printing Machine
Blick 4-Color Garment Printing Machine
Dynamic 4-Color Screen Printer
Jennings Professional 1-Color Press

Build your own screen printing machine for t-shirts!
If you want to make your own screen printing machine, I have published free plans online for how to build a simple single color screen printing press.

You will also find new and used screen printing machines on eBay: