Rather than just relying on flat color, by mixing color with the variety of textures and patterns you can catch the eyes of viewers and make your graphics really stand out. At the most basic level, a texture is any kind of graphic which depicts a multi-toned image which is not strictly representational of anything more than the variations in surface color of some sort of object. In design terms a pattern is a texture or a graphic which can be repeated endlessly. You can align multiple pattern tiles and create a larger continuous pattern.
Textures are most useful for giving images depth and variation.
Patterns have the added functionality of being able to be smaller and more compact and still lend their character to a graphic of any size. There are lots of ways you can use textures and patterns in graphic design, but let's focus on how they can help in web graphics. The most common uses of graphics on the web are titles and link buttons of some kind. While these can be made interesting with color and with tricks like offsets and beveling, large expanses of unbroken color, reliance on simple colors, and repetition of standard graphic techniques tends to produce graphics which don't grab the eye, and creates an overall impression of crudity. The use of textures can add essential variety to your graphics which give your web page a more sophisticated characteristic look and feel.