The hand weapon carried by Christopher Pike's crew during the Talos IV mission, a predecessor of the phaser that can be fired as a beam or set to build up to an overload.
The laser pistol, or hand laser, is the standard Starfleet sidearm seen in The Cage, the first pilot of the original series, and in the footage reused for The Menagerie. By Kirk's time it has been replaced by the phaser.
On Talos IV, Pike's landing party carries laser pistols and later brings down a large laser cannon to blast open the Talosians' lift entrance. The door appears untouched, but this proves to be an illusion; the weapons had worked all along.
In the Talosian cell, Number One sets her laser to overload, threatening to kill the captives rather than let humans be bred as slaves. This convinces the Talosians that the species is too violent and too fond of freedom to be of use to them.