(written from a Production point of view)
Referenced in an ultimately unused scene from the final draft script of TOS: "Dagger of the Mind", a Voice-O-Graph projection was a graphic that displayed sound waves and was used to analyze Human voices. It indicated how the talker had been feeling, for example stressed or fatigued. The script also referred to this type of display as a "voice print".
In the teleplay, a pair of Voice-O-Graph projections were carefully compared by Spock and McCoy, on the bridge of the USS Enterprise. From this pair of graphs, they were able to confirm that a message which the starship seemed to have recently received from its commanding officer, Captain James T. Kirk, was indeed actually from him. Unbeknown to the two investigating officers, Kirk had been subjected to treatment with a neural neutralizer at the Tantalus colony while the call had been made. The Voice-O-Graph projections contained tonal variations which suggested the captain had been under coercion, as well as showing that Kirk had meanwhile been feeling a mix of stress and fatigue.