A mainframe computer was a type of central computer or computer core.
In 2365, Commander Bruce Maddox wished to run an experimental diagnostic on Data that involved dumping his core memory into the Starbase 173 mainframe computer, though Data had reservations with the plan. (TNG: "The Measure Of A Man")
Later that year, an Iconian software transmission was deposited into a specific subsection of the USS Enterprise-D's mainframe computer, resulting in widespread malfunctions and injuries while the program attempted to work its way across all the ship's systems. (TNG: "Contagion")
When Captain Jean-Luc Picard was instructed to investigate a mysterious distress beacon in the Ficus sector, he had the computer download information on the beacon from a Starbase 73's mainframe. (TNG: "Up The Long Ladder")
In 2381 while being taken into custody, AGIMUS tried to trick the crew of the USS Cerritos into letting him connect to their mainframe computer. (LD: "Where Pleasant Fountains Lie")
In 2401, Worf, Raffi Musiker and the crew-on-the-run of the USS Titan-A boarded Daystrom Station in an attempt to access information from its mainframe computer. The away team was surprised to find the station's mainframe was Daystrom Android M-5-10, with elements of Data's personality incorporated into its security features. (PIC: "The Bounty")
In 3190, "mainframe" was one of the answers on Dr. Hirai's holographic crossword puzzle. (DIS: "Rosetta")