First things first, in the context of this article, when I use the word medium, I mean something you mix with paint to change its consistency. I'm mentioning this because a medium can also mean the type of paint, for instance acrylic or watercolor. (You can usually judge what is meant by the context in which the word is used.)
Texture medium (or gel or paste) is, as the name suggests, used to add surface texture to a painting. It's stiffer than paint straight from the tube, so will hold a form or shape more readily. It's also cheaper than paint, so an economical way to build up thick layers of impasto. You can mix it with a color, or paint over it.
The photo shows a tub of texture gel where I've scooped out a lump with a palette knife. You can see how the medium holds its shape. It doesn't drip or droop, but stays put. You can create peaks and grooves with a palette knife, brushmarks with a coarse-haired brush, press patterns into it, use it as glue to add collage items. It's extremely versatile!
If you're wondering about the texture medium being white rather than clear, this is one of the properties of texture medium you should pay attention to on the label...