Render engines can be real time or not. Real time engines like DirectX and OpenGL are optimized to generate as many frames as possible per second and frequently take advantage of graphics acceleration hardware. Advanced renderers that are not real time perform hardcore calculations for ray tracing, volumetrics, radiosity, caustics, etc. Think of the Final Fantasy cut scenes and the movie. That was not done in real time. All renderers take 3D geometry and material data and create 2D images.
Texturing is the art of putting skin on naked models.

Both kinds of rendering use textures. Advanced renderers tend to use basic textures (sometimes with fine detail, but still nothing special is done to them) and rely on the engine to provide detail such as bluring, lighting, shading, and special effects such as bump mapping. Real time renderers can not afford to waste all that time on special effects. So, they tend to use 'cooked' textures that have all the special effects pre-rendered to the texture. When you look at a Quake skin, effects such as highlighting and shadow are already placed there by the texture artist.
That's it a nutshell. Now technology is always advancing and special effects that used to be just in the realm of advanced rendering engines is moving to video card hardware. Even DBP has some of these features, but it is up to DirectX to determine if your graphics card can support them.
You may have heard shaders mentioned. They are a hybrid between textures and rendering. They are literally small programs that allow you to alter the rendering of a scene.
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TAZ