This is our current understanding of how Ranged and Melee Powers work.
Melee power is a bonus applied to acts that require direct contact between the player and the target, such as a sword (melee weapon). Ranged power is a bonus applied to acts that require indirect contact between player and the target, such as a gun (ranged weapon).
Applying Powers to the WeaponsEdit
Both, Melee and Ranged Powers increase the Damage per second (DPS) on matching weapons. The rate for both is one DPS point per 10 units of power, NOT one point per actual damage number.
For example, a Gun is a ranged weapon, so any increase to Range power, caused by any equipped weapons/items will affect the DPS of that weapon. 200 Range Power equals 20 DPS points.
Remember, your character also has hidden Base statistics for each of these powers, depending upon what level and what class your character is.
Other uses for PowersEdit
Doctor's use Melee power to assist their Group Heal, and Ranged power to assist their Quick Heals. (A specific formula can be found on the Doctor page.)
It is suspected that the powers may also create a similar boost for Bodyguard aggro skills. Unfortunately, there's no way to know if there's a boosting effect, because ultimately there are no aggro numbers to look at, and a lot of variables that affect aggro, so this remains merely a suspicion.
If that's true about Doctors and Bodyguards, it also possibly affects Hunter skills as well, but again, there's no way to know without numbers to show us.