It comes under the general heading of 'Finite Difference Tables'.
The example that you give is of the function for in steps of 1, (which is normally written as ).
So long as the step length is constant, (and it needn't be an integer), any quartic will have a constant fourth difference column. So for example (taken at random), tabulated for will have a constant fourth difference column. The method is linked to differential calculus, try differentiating your example and my example four times to arrive at a constant and see how it fits in with the constant difference in the fourth column.
As you might guess, any linear function tabulated at equal steps will have a constant first difference column, a quadratic has a constant second difference column and so on.
Finite difference tables can be used for Interpolation, Extrapolation (with caution), Numerical Differentiation (again with caution) and Numerical Integration.