Powers represent various things that the character can do that aren't universal. If you don't have Pattern in Amber, you can't walk between worlds, regardless of your attributes. In a fantasy setting, “powers” might include sorcery, turning to stone, or secret fighting techniques. In the Nano-Victorian Future, exclusive access to certain kinds of nanotech, special education in obscure lore or actual cybernetic modifications might be more appropriate. In a modern world with no magic or superpowers, “powers” might be uncommon skills (gun training, chemistry, etc.)

For figuring out how powers work, there's again two approaches; Amber DRPG uses the cost of a power as a sort of cost of admission. Buying a power gives you the ability to do the stuff the description says you can. How well you can do that depends on your attributes.

I wonder if that's an overly blunt instrument for something like “wizardry” in a fantasy setting though; the genre calls for a wide variation in the power of wizards, and attributes might not be the best way to achieve that. As an alternative, one could just have a power as another place to pool points, making powers a sort of secondary attribute that not everyone ranks in.

A third path might be to have “milestones” where a certain amount of investment in a power gives a character certain abilities with it, while allowing fine tuning in between.