How do textures work? Well a texture is just a grid of pixels. If your refering to texture mapping, the actual process of appling a texture to a set of triangles / model, then its pretty simple;
Basically each vertex in a model has a UV value set to it. This basically defines a point within the texture (top-left of a texture is UV 0,0 bottom-right is UV 1,1). When a triangle is rendered using the vertexs, the rasteriser basically takes the UV coordinates of all 3 vertexs, and applies that segment of the texture to the triangles face.
Thats pretty much it, nothing complex.