Electrolytic cell is the cell in which electrolysis of an electrolyte is done i.e., electrical energy is used to produce chemical energy. Whereas, Electrolysis is used to break chemical bonds via a catalyst in a solution. This converts some electrical energy into potential chemical energy. Electrical energy does not directly convert to chemical energy. They interact through several processes, electrolysis being one of the most common.
When you connect wire or something conductive between the positive and negative terminals of the battery, the electrons from the negative side travel to the positive side through the wire. The movement of electrons is called current. This is one real life application of conversions of Chemical Energy to Electrical Energy.