In a nutshell, Alpha brainwaves relate to creativity and daydreaming, Beta waves are produced in the middle of deep thinking, Delta/Theta waves can be found during deep sleep, and Gamma waves are associated with problem solving, happiness, and compassion.
Each brainwave has its associated use and helps support the state of mind required for the task at hand. All of us display each one of these waves, but at a different proportion. Depending on your psychological traits and abilities, you could have a predominant brainwave or two.
Very creative people have a predominance of Alpha waves. As we grow older, the rate of Alpha waves reduces while Beta waves increase in frequency. More Beta brainwaves relate to intelligence, and problem solving capacity, as our lives demand more of this type of thinking. But we also need Delta waves to sleep, restore and repair the wear and tear we put on our brains each day. On the flip side, we need to engage our Gamma waves to feel engaged, happy, and content.