Thinking about dimensions other than the three we're used to can rattle one's mind. That's why it's usually left to stoned conversationalists and theoretical physicists. To help the rest of us navigate flatland, fractal landscapes, and hyperspace, New Scientist put together a concise and fun tour titled "Beyond Space and Time." From New Scientist (spacetime curvature illustration from Wikimedia Commons):
What is a dimension?"Beyond space and time: Fractals, hyperspace and more"
The most intuitive description is the oldest one: the number of dimensions a system possesses is the number of independent directions you or anything else can move in. Up and down count as only one dimension because up-ness and down-ness are two sides of the same coin: the further up you go, the less down you are. The same connection exists between left and right, and forwards and backwards, but not between up and right, down and backwards, and so on. Thus, the geometers of Ancient Greece recognised, we live in a three-dimensional world.
So far, so simple, but then things start to unravel. Our place in the cosmos is defined as much by time as it is by space. As long ago as the late 18th century, the Frenchmen Jean le Rond d'Alembert and Joseph-Louis Lagrange recognised that the mathematical language needed to address time was very similar to that which described space. Time, the mathematicians of the day rapidly came to agree, was a fourth dimension.
That opened the floodgates. Once untethered from its origins in physical space, the concept of a dimension began to lose its focus. It came to be used as a general term to describe the number of independent coordinates or variables needed to determine the state of any object.