In the first Dark Forces, it was said to be just a pistol that Kyle Katarn was fond of, and your starting weapon in the game.
In later games, it was explained that the Bryar Pistol was actually the Bryar Rifle, a sought after gun by blaster aficionados, with the muzzle and stock sawed off. It was valued for it's accuracy and power and was a favorite of Katarn's, that is, until he received his first lightsaber.
In the first Dark Forces, it only fired single shots and had no secondary fire. By Jedi Outcast, the pistol received a makeover, and could now, in addition to firing single shots, be charged to fire a larger and more powerful burst.