Blender 3D: Noob to Pro/Curve and Path Modeling

Why Use Curves?
You’ve learned how versatile mesh objects can be, and how they can easily produce flat surfaces with sharp edges and even give a convincing representation of curved surfaces with rounder edges. So why do we need a separate kind of curve object at all?

There are important reasons.
 * The various kinds of curves and surfaces/patches (Bézier, NURBS etc) were used in computer graphics before meshes were developed. In the days when memory was more expensive, they offered a more compact way to represent complex shapes. Even now, they are still common in CAD and other technically-oriented graphics applications, where they make it easy to precisely specify the shape of a curve. Thus, if you are importing data from such applications, you will need the ability to represent such curves—even if you end up converting them to meshes before applying materials and textures for rendering.
 * You can use curves as guides for shaping meshes. This makes it easier to construct certain kinds of complicated curved shapes. Particularly since it is simpler to make changes to the curve (with fewer control points), than after it has been converted to a mesh. To this end, Blender defines scaling radius and tilt angle settings for curve control points, which make no difference to the appearance of the curve itself, but have an effect when it is used to deform another shape.

Bézier Versus NURBS
In the following pages, you will come across two kinds of curves/patches.

Bézier objects only occur in Blender as curves, while NURBS (“Non-Uniform Rational B-Spline) objects can be curves or surfaces. If you’ve used 2D drawing programs like Inkscape or Adobe Illustrator, you would have come across Bézier curves before. NURBS curves are a mathematical generalization of these, which are heavily used in CAD applications. Unlike Bézier curves, NURBS curves allow the specification of a variable weight for each control point, which governs how closely the curve passes to that point.

More Than You Wanted To Know About Curves
As usual, Wikipedia has the lowdown on all the interesting trivia about
 * Bézier curves
 * NURBS curves