A point or line that, when moved along a certain path (called the directrix), creates a new shape.
Also called a generator.
For example, a line moving in a circle traces out a cylinder. In the applet above, press "Run". The generatrix is the line on the right. As it moves at a fixed distance from the left line, it traces out a right circular cylinder. The path it follows is a circle. The path followed in the way is called the directrix.
In another example, a point moving at a fixed distance from a central point will trace out a circle. The moving point is the generatrix (or generator) of the circle.