A cutter beam was a hand-held tool that emitted a beam of directed energy which was used to perform a cutting function.
A cutter beam was generated by a phaser-like tool used by Starfleet engineers in the 2260s. An engineer used such beam to free Scotty from the jammed engineering core hatch in 2269.
Following his released, it was found that all the cutter beams aboard were drained of all energy, caused by an ancient magnetic organism. (TAS: "Beyond the Farthest Star")
The cutter beam was described in the script notes the tool to actually be "a cutter tool." When in use, it was explained to emit "
a phaser-like beam (different color). The beam deepens in color."