A rational b-spline is similar to a non-rational b-spline with the following exception. A "Weight" is added to each of the control points. This weight usually ranges from 0.0 to 1.0 and reflects how much the particular control point affects the curve. A b-spline is actually a rational b-spline with all weight equal to 1.0.
Modifying the weight of a rational b-spline:
When you select "Show Rational B-Spline", you will notice that at each control point, a horizontal line has been drawn. This is a visual depiction of the weight of that point. All weights start at 1.0. To modify the weight of a point, select "Move Point" on the "Curve Editor:" panel. Then "grab" a point by pressing down the left mouse button and dragging. By moving the mouse cursor over the "end" of the weight tail, and "grabbing" it, you can modify the weight.
Moving the line to the right increases the weight, while moving it to the left decreases the weight. Important: If you move left past the point, the weight will become negative. This causes very strange (and miraculous) behavior. Try experimenting with it.
Notice that when you make the weight very large, the curve is drawn to the point. When you make it very small (by moving the weight tail very near the data point), the control point nolonger affects the curve. (I.e.: It has a weight of zero.)
And the drum roll please... You are finally going to learn the hidden truth about that scary acronym: NURBs (Non-Uniform Rational B-Spline). By now you should have all the pieces to figure this one out. A Nurb is a rational b-spline with a non-uniform knot vector.
The definition of a rational b-spline can be found on page 204 of "Computer Aided Geometric Design", Cohen, Riesenfeld, 1989.