Utilizing renewable energy sources for hydrogen production represents a key step forward in achieving the goals of the hydrogen economy and the successful implementation of hydrogen as an alternative fuel for the future. Its efficient production is necessary in order to be able to compete with the fossil fuel market, whose reserves are slowly running out. One possible, efficient way to produce hydrogen is with water electrolysis using a PEM electrolyzer. In the introductory part of this paper, a brief description of the electrolysis process is given and the basic thermodynamic laws are described, which we use to define changes in the state of substances in the electrolysis process. A brief introduction to electrochemistry is given, where physical micro processes within the electrochemical reaction are described as is their influence on the required activation energy for the successful conduct of a chemical reaction. The third chapter describes the design of the electrolyzer, the necessary components and their characteristics, as well as the entire system needed for a proper electrolyzer operation and storing of products, hydrogen and oxygen. Furthermore, for the given electrolyzer, a mathematical model was derived whose purpose is to better describe and predict the behavior of the system and its output parameters. A brief overview of some of the assumptions introduced is given in order to simplify the model. The influence of some of these assumptions on the overall error in the mathematical model is also described. In the fifth chapter, one of the possible designs of the electrolyzer is modeled, with technical drawings and a 3D model made in the Solidworks software package. Finally, a simulation was made for the designed electrolyzer in the Matlab software package, which, with given parameters, is used to calculate and plot the desired output parameters and compare these results with the data obtained from several manufacturers.