You could consider this a counterpart to emulsifying. This process is a process by which the viscosity of a liquid mixture is increased. To be clear, this is distinct from emulsifying. Emulsifiers are used to combine compounds that would otherwise not form homogeneous compounds. Thickening is employed when wanting to thicken a homogeneous mixture.
When talking about thickening, it is virtually impossible to not mention molecular gastronomy. Think of thickening as one of the more basic techniques that take center stage in molecular gastronomy and its focus on manipulating textures in food (among other components). Common thickening agents include gelatin, agar agar, tapioca maltodextrin, xanthan gum, corn starch, and more.
The thickener used may vary depending on your desired effect. For example, xanthan gum may be better suited to being a stabilizer in liquid based compounds while gelatin, as the name implies, is better suited to forming different types of gels. Likewise, you may want to choose one thickening agent over another may be depending on dietary needs or regional availability. If you’re vegetarian, then you probably do not want to choose gelatin as the thickener but may want to go with agar agar. Additionally, if you are cooking foods with more Asiatic influences, particularly Japanese or Thai foods (obviously not limited to those), you may want to go with agar agar then.
Thickeners are used both for savory dishes and for desserts. Feel free to get creative with what you use and why. Just remember that a little goes a long way.