Texturing

"Put wallpaper onto 3D geometry"
Control color, glossiness, transparency, ... (demo with Cinema 4D)
How to place textures on surfaces?

Simple solution: use projections such as orthogonal, cylindrical or spherical (demo with Cinema 4D)
Problem 1: Textures slip under deformations.
Problem 2: Placement is too strongly restricted.
Better solution: "Texture coordinates". Equip every vertex with the coordinates of its corresponding point in the texture. These coordinates are called u and v, sometimes r and s. Demo with Cinema 4D / BodyPaint 3D and DirectX. Between the vertices, texture coordinates are interpolated bilinearly. On top of that, a perspective correction is applied.

Technical term: Pixels of a texture are called "texels" (texture elements).

Texture Interpolation, Texture Filtering

On rendering, the texels neither are not of the same size as the pixels of the screen, nor are the texels aligned to the pixels. Hence, we need color values "between" existing texels. Typically, these are computed using linear interpolation (or rather bilinear interpolation because it's done in two directions: u and v). Demo with DirectX.

If the texels become significantly smaller than the pixels, we'll see moiré (=aliasing) patterns in the rendering. Demo with DirectX and Cinema 4D. To suppress this effect, "MIP mapping" is used (MIP stands for "multum in parvo": much in little). A MIP map contains the texture image in several levels of resolution, each one two times the sidelength of the next. With linear MIP map interpolation, the graphics chip chooses the two most appropriate levels and blends them linearly. Hence, trilinear interpolation, because we're interpolating in u, v, and MIP level. Demo with DirectX.

Alpha Blending

We may use a forth color component (alpha, hence ARGB) to control transparency. Demo with DirectX. Note that "alpha blending" incurs dependency on the order in which objects are rendered.

Bump Mapping

Textures may also be used to control shape:

Displacement Mapping: Deform the geometry along the normal direction, controlled by a gray-scale image. Needs finely tesselated geometry. Demo with Cinema 4D. Matrox graphics cards and the newest Nvidia graphics cards can do this in hardware, too.
Bump Mapping: Do not actually deform geometry, but compute the lighting with a deform normal vector. Works with coarse geometry and hence is fast. But the profiles of objects look undeformed, and strong deformations tend to look wrong. Most current graphics cards can do this (with some programming effort). Demo with Nvidia FX Composer.