Image moments
One approach is to calculate the first and second order central moments of the shape described by the 2D contour. Using these values the elongation of the object can be calculated.
The central image moments can be combined to the seven moments of Hu, which are invariant to change in scale, rotation and translation (ie. they are very good for basic shape recognition). (More on image moments here).
Unitless ratio of perimeter and area
An other approach is to calculate the length of the perimeter (p) and the size of the inscribed area (a). Using these two values, the following ratio can be computed:
ratio = p^2 / (4 * pi * a)
The closer this ratio is to one, the more circle like is the described shape.
Other methods