A data ghost was a digital impression of a living consciousness that sometimes remained in the neural relay of a 51st century communications systems after its occupant had died. It was very disturbing to observe a data ghost; Donna Noble called it the most horrible thing she had ever experienced.
The dead person would appear to be alive and able to communicate, though confused. This was due to the degradation of the stored data pattern. They would repeat themselves or make illogical statements until the pattern failed. The Tenth Doctor called a data ghost a footprint on the beach with the tide coming in. The pattern would continue to fade until someone manually disconnected the unit.
A data ghost would communicate through the speaker on the suits, horrifying the witnesses to their death. Three of Mr Lux's exploration crew experienced the data ghost phenomenon on their deaths in the Library: Miss Evangelista, Proper Dave (TV: Silence in the Library) and Other Dave.
A data ghost could be uploaded onto a hard drive and cheat death. The dead person would live inside a computer. The Doctor did this to save River Song, who died but whose consciousness was briefly retained in a specially designed neural relay the Doctor had installed in a sonic screwdriver he had given her. (TV: Forest of the Dead)
Through an unknown method, the data ghost of River Song was capable of entering a psychic "conference call". By "keeping the line open" with Clara, River's data ghost was even capable of manifesting as a ghostly figure to Clara on the alternate timeline of Trenzalore. (TV: The Name of the Doctor)