Textures are an important and fundamental aspect of real-time rendering in general, and OpenGL in particular. The use of textures within a shader opens up a huge range of possibilities. Beyond just using textures as sources of color information, they can be used for things like depth information, shading parameters, displacement maps, normal vectors, or other vertex data. The list is virtually endless. Textures are among the most widely used tools for advanced effects in OpenGL programs, and that isn't likely to change anytime soon.
In OpenGL 4, we now have the ability to read and write to memory via buffer textures, shader storage buffer objects, and image textures (image load/store). This further muddies the waters of what exactly defines a texture. In general, we might just think of it as a buffer of data that may or may not contain an image.