The Stare is a Tier 2 Action Skill Upgrade in Gaige's Little Big Trouble skill tree, which gives Deathtrap the ability to shoot a powerful heat-beam from his eye.
Every so often, Deathtrap will pause, and fire the beam, deals constant fire elemental damage to a single enemy; additionally, this beam can pass through solid objects and thus will never miss once it has started.
Despite the description, Deathtrap actually only targets a single enemy with the beam rather than sweeping across the field.
With investments in both Buck Up and The Stare, Deathtrap will also attack Gaige with The Stare. The Stare won't do any damage to Gaige, but Deathtrap will also waste time while executing this move.
Neither Badass elemental damage bonuses nor Evil Enchantress will increase the heat beam's inflicted damage.
The skill's name, like several others in Gaige's repertoire, is a reference to the animated television show My Little Pony: Friendship is Magic. One of the series' main characters (Fluttershy) possesses an accusing gaze that can paralyze creatures with fear.