A deposition was an account given by a person into an investigation.
In 2267, after the apparent death of Ben Finney aboard the USS Enterprise, Captain Kirk gave a sworn deposition that he had jettisoned an ion pod with Finney onboard only after the ship went to red alert. Since the ship's log said otherwise, Commodore Stone accused Kirk of willful perjury. (TOS: "Court Martial")
In 2366, the destruction of the Tanuga IV research station and the death of Dr. Nel Apgar in a mysterious explosion was investigated by the Tanuga IV security force and the USS Enterprise-D. As part of the investigation, the parties involved gave depositions as to what happened in the events leading up to the explosion that were used to create holographic recreations. Commander William T. Riker's deposition was Deposition Program Riker One. Dr. Apgar's wife, Manua Apgar, used Manua Simulation One as her deposition while Dr. Apgar's assistant, Tayna used Tayna Simulation Three and Tayna Simulation Four for her depositions. (TNG: "A Matter of Perspective")