Regarding where the game takes place (modern world / future / fantasy etc.) - this is just called the setting of the game, e.g. "first person shooter in underwater punk setting".
Regarding the gameplay - games are sometimes called "realistic" - where there are realistic limitations on the action and the world - versus "old school" - where you can (for example) run indefinitely, survive many shots, etc.
I haven't heard any term in particular being used to describe games in which you have special "powers" vs games where you just shoot your enemies in the face.