For each color component channel (RGB) or the composite (C), this node allows you to define a bezier curve that varies the input (across the bottom, or x-axis) to produce an output value (the y-axis). By default, it is a straight line with a constant slope, so that .5 along the x-axis results in a .5 y-axis output. Click and drag along the curve to create a control point and to change the curve's shape. Use the X to delete the selected (white) point.
Clicking on each C R G B component displays the curve for that channel. For example, making the composite curve flatter (by clicking and dragging the left-hand point of the curve up) means that a little amount of color will result in a lot more color (a higher Y value). Effectively, this bolsters the faint details while reducing overall contrast. You can also set a curve just for the red, and for example, set the curve so that a little red does not show at all, but a lot of red does.
Here are some common curves you can use to achieve desired effects: