Every 3D scene has some kind of structure in it, which is modeled from a plan, design or just an idea. This structure is made of a virtual set of lines and faces that can be faceted enough to produce curves and seemingly organic shapes.
You can see by the scene above that everything is just lines on a gray background, but if you move your mouse over the image it will switch to the final 3D rendering. The 3D artist has many tools to aid in making 3D Mesh, such as splines, NURBS and other weird sounding things, but it all comes back to tiny triangles, and more recently squares, that form the structure of a scene, object or character in 3D.
There are also different kinds of software for making mesh, each a specialist in it's own right. For example you may want to use ArchiCAD to create a building but Rhino 3D to make organic features. Each can export the mesh into another application where the 3D Artist can prepare and create renderings and animations.