Informally, the dimension of a shape is the number of numbers that is needed to describe the position of any point in it. For example, a cube is three-dimensional. Three numbers are needed to describe a location in a cube - an x coordinate, a y coordinate, and a z coordinate.
Another definition of dimension is based on "roughness". Generally, the dimension of a shape is logscale factor(factor by which the overall space that that shape takes up increases).