The process used by plants to release oxygen is called photosynthesis. The plants convert carbon dioxide and water into carbohydrates and oxygen using the energy of the sunlight. This process happens during the day because sunlight is available only during the day. Mostly, people think that plants breathe in carbon dioxide and give out oxygen. However, it is not so.
Plants also need to break down carbohydrates into energy for which they require oxygen. Now, you might ask why do plants release all the oxygen produced during photosynthesis. The answer is plants do not give away all oxygen as they retain small amount of oxygen that they use to break down carbohydrates to generate energy. In the night, plants breathe oxygen from the air and release carbon dioxide as animals do.
This much is about how plants produce oxygen. On the other hand, there are oxygen plants which are used for producing oxygen commercially. An oxygen plant is industrial gas equipment for generating artificial oxygen. An oxygen plant manufacturer might use either cryogenic distillation technique or pressure swing adsorption method for producing oxygen.