A piecewise cubic hermite curve is a curve that is represented with four degrees of freedom. Two degrees of freedom are defined as the positions of the two end points of the curve. The other two degrees of freedom are defined as the tangents to the endpoints of the curve. Pictorially this is displayed on the "Curve Editor:" canvas with the beginning control point of the hermite curve displayed in red, and the end control point displayed in green. The tangent of the beginning point is in blue, and the tangent of the end point is in pink. A given point in time of along this parametric curve is displayed by a white dot on the "Curve Editor" canvas and by a white vertical line on the hermite basis functions canvas.
All points along a hermite curve are represented by a given time (ranging from 0.0 to 1.0). Each point in time is determined by adding the red basis function's value multiplied by the red control point, plus the green basis function's value multiplied by the green control point, plus the blue basis function's value multiplied by the blue tangent, pluse the pink basis function's value multiplied by the pink tangent.
The curve is called a piecewise curve because it is created/drawn one piece at a time. This is the reason that all but two of the control points of the curve are displayed in gray. The basis functions are repeated between each of the sets of two points on the curve. Notice that if you have a hermite curve with three control points, that moving control point 0 (or its tangent) has no affect on the curve between control point 1 and 2.