Powerups are incredibly useful and serve many different purposes. Each powerup has a very specific use, however the situation in which you use them can vary extremely. You can find powerups all over the place, but they can be found especially often in those hard-to-reach places.
There are three different powerups: Fireball, Shield, and Restoration. Each powerup has its own unique rarity, but they are all relatively rare; you should use them sparingly and only when you really need them. Activate power ups by tapping the icon when they appear in the upper right. If it will keep you from taking a hit, that is always a good use, but for the most part they are meant for getting you out of dire situations.