As you know, a catalyst is a substance that alters the rate of a reaction by decreasing the activation barrier. Inorganic catalysts are usually composed of metals( for instance Fe in Haber Process and V in Ostwald process) which mainly employs the idea of providing more surface area and thereby decreasing the activation of energy for a reaction. Organic catalysts in the other hand is composed mostly of organic molecules like amino acids (Enzymes) and Nucleic acids (eg: Ribosome. In ribosomes where amino acids join together to form proteins, the active site, where the reaction take place is composed of RNA molecules). The major difference between inorganic catalysts and organic catalysts are organic catalysts are stereospecific, that is, enzymes that could catalyze a reaction might not be able to catalyze a reaction with a different substrate (eg : Hexokinase, an enzyme that adds phosphate group to C6 carbon of glucose can't add a phosphate group to fructose, which is an isomer of glucose) while most inorganic catalysts are not stereo specific.
Hope this helps.
