The TR-116 is a specialised weapon developed by Starfleet for use in areas where normal phasers would be useless - within dampening fields or radiogenic environments, for example.
The weapon was designed to be as simple and foolproof as possible - it used a chemical explosive to fire a Tritanium bullet and had no electrical or optical systems at all. The performance was poor by the standard of phaser weapons - range was limited to around one kilometre at most, and the largest feasible magazine was only capable of carrying thirty or forty rounds. However, in the kinds of environments the rifle was designed to operate in most opponents would be completely unarmed, while species like the Klingons would be reliant on swords and knives. Against that kind of opposition the TR-116 was more than adequate.  The TR-116 could be operated by both right and left-handed shooters. 
The development of regenerative phasers which can also operate within energy-hostile environments made the TR-116 obsolete, and Starfleet dropped the program as soon as they were confident of its replacement. For some years the design remained merely a forgotten replicator pattern, but in 2375 a modified TR-116 was used to commit three murders on board Deep Space Nine,  when Science Officer Chu'lak used the weapon to murder fellow Starfleet officers Hector Ilario, Greta Vanderweg, and Zim Brott. 
The altered weapon included two major changes. First, it was fitted with a microtransporter; when the bullet was fired the transporter beamed it to within less than 10 centimetres of the target. By using an exographic targeting sensor the killer was able to scan through many layers of bulkheads, allowing the TR-116 to be fired through walls or flooring. Chief O'Brien reproduced these alterations on another TR-116, which Lieutenant Dax subsequently used to find and capture the murderer. Operating the weapon is simplicity itself; the targeting scanner is located on the rifle, transmitting its viewpoint to a headset worn by the user. A simple thumb control moves the viewpoint forwards and backwards, allowing it to pass through walls as needed. The targeting graphic cues the user to fire whenever a target is in the line of fire.
Some thought has been given to producing the modified TR-116 as a field weapon, but while the displaced targeting system is ingenious the basic limitations of a projectile weapon remain. Since phaser beams can be transported on the way to the target much as bullets can, displaced firing is likely to become a feature of phaser weapons in the future.
In the non-canon Deep Space Nine Mission Gamma novel Lesser Evil, Lieutenant Sam Bowers tells of his experiences aboard the USS Budapest during the Battle of Sector 001. The Budapest was one of a handful of Starfleet vessels equipped with prototype TR-116s. When the Borg attempted to assimilate the ship and its crew, Bowers led a security team equipped with TR-116 rifles through the ship, with orders to eliminate any and all Borg drones. Because the weapons used projectiles instead of energy beams, the drone's deflector shielding was useless, and the Budapest crew were successful in stopping the Borg.
Though these events were never mentioned in "Field of Fire", the statement about projectile weapons being effective against the Borg is supported in part by the scene in Star Trek: First Contact where Jean-Luc Picard uses a holographic Thompson submachine gun to kill two Borg drones. It is unlikely that Borg have any kind of kinetic energy shield, as there have been dozens of incidents in which Borg drones have been killed by blows from rifle butts or bat'leths multiple times, and no adaptation was seen.
In the novel A Time to Heal, a Starfleet security sniper used a TR-116 to stun Tezwan guards. It was noted that the gun could "shoot through walls", indicating that Chu'lak's modifications may have been adapted for field use.
The fact that projectile weapons have not become standard-issue is probably a strategic decision by Starfleet; the limited environments and adversaries for which projectile weapons are an advantage over phasers are not sufficient in number or threat to justify the logistical problems of stocking or replicating ammunition. It may also be that the TR-116 is too close to being a strictly lethal weapon to comfortably issue, whereas phasers can function as tools in addition to its use as a weapon.
Behind the scenesEdit
John Eaves has stated that it uses a .30 caliber round. 
In the novel Star Trek: Destiny - Lost Souls, David Mack describes the TR-116 as using 10mm Monotanium slugs. This addition of information could explain that the rifle comes in several different calibers (like many modern firearms) or that it itself can be altered to fire different types of ammunition through a function on the rifle or a quick-conversion/part-swap kit.
The TR-116 rifle is available for use in Star Trek Online free from the c-store for those who preorder the game from Target.