Vertical forward grip

A vertical forward grip or foregrip is a vertical pistol grip mounted on the fore-end of a long-barrel firearm, designed for grasping by the frontal support hand (or "off hand"). 
Use.
Forward grips aid in the maneuverability of the firearm, since the natural angle of a person's outstretched hand is more oriented to grasping objects at a vertical angle, rather than a horizontal one perpendicular to the body. Foregrips can decrease accuracy in precision rifle shooting if the shooter tends to "muscle the weapon".
Vertical foregrips can have features located inside the inner diameter, such as a deployable bipod inside the grip's housing. The M4 Carbine's with similar weapons' different Grip Pod Systems, AKA "Grip Pods" created in 2003 are still used by the American (Include the US Army, USMC, National Guard, US Air Force, FBI, CIA, DEA, Customs Border Patrol, etc), British and Australian armed forces is such a device. The grip may also have a tactical light and the control switches molded into the grip's assembly. Other designs may contain storage space for spare parts, spare batteries for optics, weapon lights, or small first aid gear. Some designs may have a bottom that threads on with an O-ring to make this storage compartment watertight.
Stubby grip.
A shorter version is referred to as a "stubby grip" and is primarily intended to keep the shooter's hand from moving forward of the muzzle.
Angled forward grip.
Angled forward grips can provide similar functionality to vertical forward grips. One such popular product is the MagPul Dynamics AFG, or Angled Fore Grip. This allows the operator to lay the palm of their hand onto the grip while pushing the thumb down on the top of the gun.
Legality on pistols in the U.S..
The legality of vertical foregrips on pistols in the United States is unclear. In the United States, firearms are categorized by the National Firearms Act and firearms manufactured with specific features are subject to restrictions and controls. In the case of a vertical forward grip on a pistol, the law does not explicitly define such a feature as belonging to any category.
In May 1993, in response to legal action brought by the Bureau of Alcohol, Tobacco, and Firearms (ATF), the South Carolina District Court's finding of fact concluded that a pistol modified with the addition of vertical fore grips was still a pistol and not any other weapon device. Following this, the ATF dropped their charges and the case was not tried; as such no precedent was set.
In an open letter sent to Federal Firearms Licensees in April 2006, the ATF stated their interpretation of the law: that installing a vertical forward grip on a handgun is the same as manufacturing an Any Other Weapon (AOW) category firearm and subject to registration and taxation, with significant penalties for manufacturing or possessing such an unregistered weapon.
The ATF has stated that angled grips do not make an AOW from a handgun.