The Boolean gun was a type of large hand-weapon. It did not use projectiles. When activated, it produced a loud roaring noise and anything it targeted simply disappeared, rather than being burnt through.
It was J.J. Abrams who devised this type weapon. Recalling how he did so, Ben Grossman stated, "J.J. one day ran up on set and said he had this great idea for this new gun. Imagine if there was a gun – a really bad-ass gun – [...] that's like a Gatling gun and when it goes off it's kind of like the sound of God clearing its throat – just like 'raaaaaaaaaahh'. The 'boolean gun' development and execution was spearheaded by Look Dev Supervisor Max Reiss." 
To depict the gun obliterating Klingons and D4-class Klingon fighters, Pixomondo suggested stitching both live action and planned CGI shots. "So we mixed in the work of the special effects team and we did CG takeovers for a lot of that stuff," said Ben Grossman.