Powers are abilities that are supernatural or extraordinary in nature that make you...well, more powerful.
Spells are an obvious type of power, but there are others, such as passive or constant effects, supernatural objects or gear, or even the effect of your legacy.
Mechanically, powers are simple: you gain 1 per level, plus 1 at character creation (i.e. level "0"). You can select any power as long as you meet any specific requirements it calls out.
For convenience, powers are organized into groups. These tend to describe powers coming from the same source (natch), and some rules that tend to apply to all powers in that category. Examples: