Well hey, I took serigraphy I and II in university (ie, screen printing) and boy, it is very, very different from a digital printed poster. Both Kip and nostalgicroxas give some great insight!
Paper printing with serigraphy is fairly common -- serigraph paper like somerset is recommended but you can absolutely print on all types of paper. I printed on rice paper, vinyl sticker, and watercolour paper with surprising ease (the vinyl sticker paper didn't quite agree with serigraph inks but it worked!). Below is the rice paper dragon pinata I made for a show -- screen printed on rice paper.
The process is a traditional edition printing method now more closely associated with t-shirts but it's roots in paper go back very, very far. The complexity of a poster printed via serigraph depends on the number of colours used in a print. Each colour will have its own area blocked out and burned into a screen with photo-sensitive emulsion fluid (as seen in kip's attached photo) which is then transferred to paper via inked squeedgee pulls.
Each new colour requires tight registration otherwise the colour printed will be "off" (think of the hairline whitespace or gaps between the lineart and printed colours often seen in newspaper comics). The paper must be arranged in the same exact place down to the decimal point under the screen in order to produce consistent, well-aligned prints.
Here's the screen printed stickers I made on vinyl. Very classy, edition of 40.
It's very time-consuming, VERY labourious, and requires an immense amount of careful planning. Ink goes everywhere and finding a pristine print with ink only where ink should be is like a diamond on a freeway.