They've got it all laid out on their site, but the main thing is they only accept submissions through their online manager, Submittable. No email, no paper manuscripts mailed to the office. You'll need to create an account there. The reading periods aren't year-round either; poetry, fiction, and nonfiction have specific windows they announce, so you have to watch for those. I missed the last fiction window by a week and had to wait months for the next one. It's a bit of a patience game.
They're clear about wanting previously unpublished work. Simultaneous submissions are okay, which is a relief, but you have to withdraw your piece immediately if it gets accepted elsewhere. The formatting is pretty standard: double-spaced, readable font. No crazy experimental formatting in the document itself, I'd guess. The response time can be long, like six months or more sometimes. I submitted a short piece last fall and just heard back a 'no' this spring. It's not a market for quick turnarounds, that's for sure.
Still, getting something in 'The Paris Review' feels like a real milestone, even just trying. The bar is astronomically high, but you can't win if you don't play, right? I just keep my expectations realistic and my submission tracker updated.