A Bezier curve in its most common form is a simple cubic equation that can be used in any number of useful ways. Originally developed by Pierre Bézier in the 1970's for CAD/CAM operations, it became the underpinnings of the entire Adobe PostScript drawing model. If you're a regular user of Adobe Illustrator, Macromedia Freehand or Fontographer, any number of spline-based 3D programs, etc., you've probably used Bézier curves.
It can also be used for animation. These pages are a simple introduction to how to implement Béziers for motion control: how to choose curves and control points, and how to calculate movement along the curve. I've provided a sample Shockwave movie you can try out, download, and grub around in the code of.