In the third book of The Hunger Games Trilogy, Mockingjay, Katniss is given a new bow designed specifically for her. The bow was created by Beetee, and the book hints that it has some special capabilities. It doesn't expand on what these capabilities are besides it humming or vibrating while active. What are this bow's abilities?
The book is a little vague, but it appears the bow was designed to dramatically increase accuracy. This is made evident when Katniss first uses the bow in combat to shot down the hover craft. Here, Katniss remarks to the reader that she is firing MUCH farther then her old bows would have allowed, and her shots are accurate (which would not be possible with a normal bow due to the physics of shooting an arrow long distances).
At first I thought the bow's special arrows used guidance mechanisms to keep the arrow on course. While this is a possibility, the fact that Katniss and Gale both missed their first shot implies that this is not the case.
I believe the more reasonable ability would be to compensate for the natural shake of the person wielding the bow. As you probably know, humans can not hold very still. This slight movement caused by blood moving though the veins, muscle shudder, balance correction, and even breathing (if the shooter isn't holding their breath) is imperceptible to most humans, but when projected out a few hundred feet, becomes quite noticeable. To alleviate this, the bow would detect the slight shake, then produce a counter shake which would result in the bow not moving.
To recap, the bow probably counters user shake and the arrows may be smart (correcting course in flight).
Update for previous comment:
Also note that when the bow is deactivated the humming/vibrating stops. I do not think the bow controlled when/if an arrow detonated. Returning to the shooting down of hovercraft, Katniss and Gale decided to start with the flame arrows first, because the fire arrows would be less dangerous if they missed their targets (and the arrows detonated on the buildings/people below).
According to this post on the Hunger Games wiki it seems that Katniss' bow in the final book is not necessarily a space-age bow. The post does not mention what materials the bow is made of, nor (to my recollection) does the book.
In Mockingjay, Katniss is given a special, military-grade bow for the rebellion in District 13. It is highly advanced, specifically crafted to fit Katniss, and uses explosive and incendiary arrows alongside regular ones. Beetee designed it to look showy on the outside, but used the inside to make it a deadly weapon, disguised as a sleek bow.
Given the tech level in District 13, I'd guess it's a combination of several things. She mentioned being able to shoot extremely accurately at 100 meters or so, which gives me the idea that the bow is able, possibly with the help of the arrow's tech, to compensate for any movement of the archer, as well as variations in the environment. Missing the first couple of shots wouldn't be too surprising if she and Gale didn't have much time to practice with their bows, especially in outdoor environments. After the bow "learned" the archer, as well as the flight of the first arrow, the accuracy would have improved. This is, of course, just a guess. I haven't quite made it through the first third of the book yet.
Since anything electronic uses power, it would be useful to have an "off" switch. While the bow is off, it could be charging with the help of the archer's movement, or some other method. The arrows would be more complex -- and simpler -- in that they'd have the mechanisms for correcting their flight path, but wouldn't need recharging so much, being nominally 'single use'.
To me it seems the bow humming and vibrations would be motors and servos working to move cams (assuming its a compound bow) and put tension on the bowstring to move dense limbs (maybe a carbon steel alloy) closer and the motors keep the string on tension thus less work on its user so more accurate shot, coupled with a scope (maybe a 4x15 power scope), lasers, and archery stabilizers would help the shooter stay on target. As for distance well the dense alloy resistant to bending would be very ridged giving the power she needs to get the distance. Her first shot being a miss is because the motors and servos had not warmed up AND she may have had a tough time pulling the string with out the motors help.