From AMS Glossary
A function composed of polynomial pieces defined on adjoining subintervals and with continuity conditions imposed on the function and its derivatives at all points including those connecting the subintervals.
The most commonly used spline function is the cubic spline that is made up of polynomial pieces of degree three, where the polynomial coefficients are determined in such a way that the composite function and its first and second derivatives are continuous at the connecting points. The graph of the cubic spline is a smooth, nonoscillatory curve. An interpolating spline is a spline function that passes through a finite set of data points.