A squat toilet (alaturka, kakkoos, washiki, etc.) is a toilet used by squatting rather than sitting. Instead of a raised bowl, the user must carefully squat over what is essentially a porcelain (or concrete or stainless steel or dirt) hole-in-the-ground to do their business. Scared? You shouldn't be. They're surprisingly easy to use and are actually pretty practical. Without going into too much detail, I will say that to properly use one, you face the wall (or flushing mechanism) and scoot as far forward as you comfortably can. For first timers and those with slight mobility issues, it is often recommended you grab hold of the plumbing at the front to prevent yourself from "falling in".
Though some would say the initial awkwardness of using a squat toilet well outweighs the practicality, I don't agree. Not only are they easier to keep clean than a bowl toilet, they are easier to repair and much more sanitary (as you're not sitting on a shared surface). There are other health benefits to squatting as well, but I'm not about to go there.
Oh, and don't underestimate the water pressure of these babies. They'll do their job just fine and do it without using as much water as a bowl toilet. How's that for practicality?
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Anyway, the idea for the squat toilet is one that most likely evolved from a unique drainage system that was created in Nara, Japan sometime between 710 and 784 A.D. Essentially, a long 4 to 6 inch wide (10 to 15 cm) drainage ditch was dug through the town which was thin enough for people to squat over with one foot on either side to easily take care of things. Lucky for you, the technology developed over time into something much more private and sanitary. So next time you see one of these "urinals-in-the-floor" put on your "I've got this" face and own it! If nothing else you'll have an interesting conversation starter for the rest of your life.
Remember, just because it's different doesn't mean it's scary. Breathe, relax and you'll be just fine.