The speech center was the section of the brain responsible for speech.
In 2268, Spock asked Doctor McCoy to finish reconnecting his speech center so that he could assist the doctor in re-attaching his brain. (TOS: "Spock's Brain")
In 2371, The Doctor suggested that Kathryn Janeway's speech center might have been damaged by spatial distortions, resulting in her babbling. (VOY: "Twisted")
In 2373, an evil version of The Doctor unblocked B'Elanna Torres's speech center so that she could tell him how to delete the original Emergency Medical Holographic program, having previously paralyzed her. (VOY: "Darkling")