Textures allow you to vary the look of a shader across a surface by mapping values to shader properties.
The basic shaders have properties that can be measured at each point on a surface: color, shininess, displacement, and so on. In a new shader these values are uniform, so for example the entire surface has one color.
To create more interesting materials, you can map a texture onto the properties of the shader. There are many different types of textures available: color ramps, checkerboard patterns, fractal noise, and more.
For example, to create a material with a smooth transition from blue to green across the surface, and that alternates between reflective and dull in a checkerboard pattern, you could map a blue-to-green ramp to the color parameter of the shader, and a checkerboard pattern to the reflectivity parameter.