Texture is tactile and visible structure on a surface. For example, fabrics have a typical texture that comes from the threads and the weaving technique.
Texture in art
In painting and drawing, the visible properties of texture are simulated using the light and shade technique at a small scale: by applying different colors (often shades and/or tints of the surface's base color) in typical patterns of dashes or dots. Even if only one color is used (as often in drawing) textures can be created by various brush/pen/pencil techniques.
Texture in computer graphics
In computer graphics, texture often refers to filling an area with a repeating pattern. For reasons of simplicity, these patterns are usually rectangular and tiled in vertical columns and horizontal rows - which limits their use to simulate texture in a picture. In 3D modeling software, textures can be 2D or 3D ("bump maps") and result in more realistic images.