A blue sigil appears on the forehead of the caster for 15 seconds.
The amount healed is typically modest compared to most healer class abilities, but don't forget about it - this ability costs no resources, and does not trigger the global cooldown (though it is affected by it).
Gift of the Naaru is a great emergency button for draenei of any class. Since it has no cast time, it can be cast on the run or in a crunch. Priests, Paladins, or Shamans can drop Gift, and then follow it up with a fast heal or two to get back on their feet.
In raid/instance situations, when the entire group takes massive damage (such as Festergut's Pungent Blight), a draenei can pop this to alleviate some of the stress on the healers.
It can be cast on other players and although it is not the most powerful heal it can help when thrown into a regular healing rotation.
Because it doesn't trigger the GCD, tanks (or pretty much anybody) can drop it on themselves to help out with the healing, and not lose too much other work.
This ability will cause shadow priests to drop [Shadowform] thus requiring a GCD to reapply it.
0600Patch 6.0.2 (14-October-2014): Heals for the same amount, but now over 5 seconds, so quicker heal. 
0400Patch 4.0.6a (11-Feb-2011): There is a maximum limit to the amount healed when used on friendly NPCs with large amounts of health.
0300Patch 3.1.0 (14-Apr-2009): The cast time has been removed; it is now an instant cast.
0200/0300Patch 3.0.2 (14-Oct-2008): The amount healed is increased based on the caster's spell power or attack power, whichever is higher. In addition, the ability is now immune against pushback effects.