In a slightly less technical (and yes, therefore less precise) explanation, proprioception is what that nice police officer who stopped you is testing when s/he asks you to spread your arms, look up, and then smoothly touch your nose with your fingertip. It is the 'sense' of knowing where your body (and its various parts) are in space, in relation to other parts of the body or to surrounding objects, without looking. There are neurological disorders that cause a person to lose this 'sense' or ability, forcing them to carefully watch what they are doing at all times so as to have visual feedback. This is probably why The Man Who Mistook his Wife for a Hat is softlinked here; it's an excellent book on such disorders.

Although this sounds like something you'd only use in a dark room, you in fact use it all the time. Touchtyping is proprioceptive. Reaching for a glass of soda while reading and picking it up by touch is proprioception. That quivery feeling on the back of the neck you get when you're walking in a dark hall and you know you're close to the wall - that's proprioception, although an extreme case.

Although there isn't an official name for it, I propose propriocarception for the ability to sense where your back bumper is when parallel parking. (wharfinger: without impact.) momomom notes that with impact, you have parking by Braille.