Living "off-the-grid" is when a person (or people) live self-sufficiently without relying on public utilities. (The utilities would be the "grid.")
Typically, these homes operate without municipal water, sewer, natural gas, or electrical power. Many homes rely on renewable energy (wind, solar, geothermal, or stand-alone), and use rain water or water from a lake or stream to provide water.
These homes are often built for economic and/or environmental consideration. In theory, it's cheaper to provide water and power without "the grid."
The term "off-the-grid" should never be confused for "off his rocker" or "out of her mind." The phrase is not synonymous for "absolutely batshit crazy" even if you think this phrase is applicable for "complete wingnut." (Even if Daryl Hannah, and Ed Begley Jr., who promote this style of living, do seem a little "nutso."*)
*I'm not saying anyone is nutso. But you might.
Off the Grid
Living Off the Grid
How Living Off the Grid Works, How Stuff Works
Stand-alone power system, Wikipedia
Off-the-Grid Homes, Design Sponge