Let A be a finite dimensional algebra and let A-mod be the category of finite dimensional A-modules. An additional structure on the algebra A often induces a structure on A-mod, and a property of A may imply a property of A-mod. Conversely, categorical structures and properties of A-mod gives structures and properties on the algebra A. In this lecture series, we consider the problem of translating between algebraic and categorical structures/properties, and we take the example of highest weight categories. Namely we will (after introducing their definitions and basics) explain the equivalence, due to Cline-Parshall-Scott, between a quasi-hereditary structure on an algebra A and a highest weight structure on A-mod. We also aim to discuss a generalization due to Kleshchev to a less finite algebra and less finite modules: the affine highest weight categories and affine quasi-hereditary algebras.