(covers information from several alternate timelines)
A grenade launcher was any of a number of projectile weapons used to propel a small bomb called a grenade. Grenades were often thrown, but a grenade launcher could be used when greater range and accuracy were required than could be achieved by a throw, or when a rocket-propelled grenade was launched. (TOS: "Arena"; ENT: "Rajiin"; Star Trek Into Darkness)
In the mid-22nd century, a Xindi grenade launcher could be used to fire, with a blue light, an organic grenade. In 2153, a Xindi-Reptilian member of a boarding party that invaded Enterprise NX-01 used a Xindi grenade launcher connected to a biorifle. During that battle, the weapon was fired at a group of Starfleet security officers who were guarding imprisoned Oran'taku native Rajiin in the ship's brig. (ENT: "Rajiin")
In 2259 of the alternate reality, Starfleet security personnel had access to automatic grenade launcher-type artillery weapons that fired small torpedo-shaped rocket-propelled grenades with aerodynamic attachments. When Khan Noonien Singh assaulted the Daystrom Conference Room at Starfleet Headquarters in San Francisco, the Starfleet personnel inside the building attempted to take out Khan's jumpship with a rocket-propelled grenade. Khan was, however, able to evade the grenade with evasive maneuvers. (Star Trek Into Darkness)
In 2267, Captain James T. Kirk found a Starfleet-issue mortar-type grenade launcher that had survived the Gorn attack on the Cestus III colony. With it, he was able to force the Gorn to decamp so that his landing party could escape their ambush. Presumably, the Gorn were not expecting to meet such determined resistance. A grenade was launched by Kirk to impact at a range of about 1200 yards; which, according to tactical officer Kelowitz, was "a little close." At that range, there was little shock wave or debris. (TOS: "Arena")
The device Kirk used in "Arena" was tube-based and would have been called a mortar by a 20th century soldier. The ammunition had no visible guidance system or aerodynamic attachments. Kirk however made adjustments for range and azimuth on the tube. This would suggest the presence of some type of a guidance system.
In the computer games Star Trek: Voyager - Elite Force and Star Trek: Elite Force II standard and enhanced compound grenade launchers (β) are available for the player. They are a heavy weapons that can fire energy grenades with time delayed detonation, 'sticky' grenades with proximity detonators, or remotely triggered grenades.