I don’t know how other photographers describe texture, but I always think of it as visually representing what you would feel if you reached into the photograph and touched the surface of the subject. By the way, don’t do this to an elephant!
Speaking of elephants, they’re one of the best subjects for showing texture in photographs due to their endlessly fascinating skin, particularly of the trunk. If you click the images to make them full size, you can see the rough texture. Making sure the texture detail stand out is a way to make the photograph more engaging and intimate.
It’s not just elephants that have interesting texture though. Following are a grey seal (feel the texture of the whiskers, the sand and the fur), a close-up of a leaf (the veins are so intricate and beg to be touched) a lioness hunting (if she wasn’t searching for breakfast, it would be lovely to run your hands through the grasses), and a crossbill in an evergreen (again, I just want to run my fingers through the soft needles)
To really home in on texture, it’s often worth cropping into the region that has most physical interest and increasing the contrast, sharpness and perhaps clarity in post-production. This brings out the edges in more detail. Often, it’s worth converting to black and white to concentrate the eye on the texture in the image.
Let’s go back to elephants for the last couple of images. They really are the kings of texture!