Wire Frame - "old" type

Wire frames are used when modelling clay and other materials (such as paper, rubber etc) that aren't strong enough to be self-sustained. Wire frames are used in art and in more "serious" modelling such as models of bridges and buildings etc.

When making proper wireframes, you usually begin by taking rather firm wire (such as 1 mm wire), and bending it so it forms the skeleton of whatever you are modelling.

From this, if your model will be large, you use thinner wires to make the shape better. Depending on your final material, you can do the model itself, or go to finer wires to make an even more nuanced model.

If you have ever seen any Wallace and Gromit movies, you have seen plenty of these wireframed figurines in action.

Wire Frame - "new" type

When working with 3D graphics, wire frames are virtual polygons that are used for modelling the final 3D models. The wireframes are used to be able to see in detail how the models work and are attached to each other.

A virtual wireframe like this looks like the 3D model, except instead of having filled polygons, it shows the borders of the polygons.

A variation of a wireframe is the lit wireframe, which essentially is the same thing, but also indicates where the highlights and shadows of the model will be when the final renderation is executed.

Wireframes were used particularily when the computer technology was not very advanced, because it took less processing power to work with the wireframes instead of with flat shaded models or simpler non-rendered models.

As the computers have become more powerful, wireframes are used less and less, and simpler renderations of the final scene are used instead. Wireframes are still used every now and then to make things look more cyberspace-ish.