There are five fundamental dimensions, quantities that can be readily measured in experiments. Constants like π have no dimensions, they are said to be dimensionless.
Dimensions aren't the same as units.
For example, the physical quantity, velocity, may be measured in units of metres per second, miles per hour etc.; but regardless of the units used, speed is always a length divided by a time, so we say that the dimensions of speed are length divided by time, or simply L/T. Similarly, the dimensions of area are L2 since area can always be calculated as a length times a length.
Dimensional analysis can be used to verify that both sides of an equation have the same dimensions. For more information, see pages in Wikipedia, at the University of Guelph, and for the more adventurous, DOE Maxima.