The number of independent directions within a set of points, alternatively: the number of coordinates needed to give each point a unique name. This is rather abstract – time for some examples:
A line is one-dimensional. There’s only one direction to go on the line (the opposite direction isn’t counted extra): Back-forth. A single number is sufficient to define a point of the line. For instance, on a motorway, given the statement “the accident happened 4 kilometres from the beginning of the I95 (or M1, or whatever)” is sufficient information for the rescue workers to know exactly where to go.
Surfaces are two-dimensional, as there are two independent directions: back-forth and left-right, say. On the earth’s surface, the two coordinate numbers geographical longitude and latitude uniquely define each location.
The space that surrounds us is three-dimensional. There are three independent directions, say back-forth, left-right and up-down. In order to define a location in space, one needs to specify three numbers – for instance, two to specify where a house is located on the earth’s surface (latitude/longitude, see above) and one floor number (or, more precisely, the height above the earth’s surface).
Adding time to the three space coordinates (a must for defining an appointment – where and when?), the result is four-dimensional spacetime. In order to define an event in spacetime, one needs to give four numbers: three of them determine where in space the event happens, the fourth gives the time where it happens.
According to some of the models that have been studied as candidates for a theory of quantum gravity, our world should have even more space dimensions than the usual three. Some information about these extra dimensions can be found in the spotlight topics “Extra dimensions, and how to hide them”, “Hunting for extra dimensions” and Extra dimensions and simplicity.