In form the Verifier is a chair with an arm that extends into a palm pad with white lights. In function the Verifier is a "perfect" lie detector using brain wave and physical clues read from the sensor on the chair arm to determine if the witness is truthful. The palm pad flashes and the computer sounds a warning if the witness is lying.
Star Trek never goes into the soical implications of such a device. Being able to find the truth quickly and without error would render a great deal of courtroom drama and maneuvering pointless. Laws against self incrimination would also have to be altered, or more rigorously enforced. When you can ask the defendant if they murdered Mr. Body in the study with the candlestick, and the truth will be known, that is a powerful and dangerous tool.
Some would argue that the truth should be known, and if someone did the deed, they should say so. Others will argue that the device obliterates the right of a person to not incriminate themselves, and defendants must not be subject to it.
At the very least the form of trials both criminal and civil will drastically change.
For form I borrow a great deal from H. Beam Piper whose universe had the Veradicator, a device with exactly the same function. Trials evolved into two stages. First the finding of fact. It was determined what happened to whom, when and where. Then came the second part of the trial, the discovery of why. The very existence of a crime often hinges on the why. If Mr. Body was minding his own business when you hit him, that is murder. But if he was threatening you with the gun, that is self defense. To see this format in action I recommend "Little Fuzzy" a good portion of the book is a court procedural in action and demonstrates at least how Mr. Piper sees the device changing things.