Rossum's beating heart
In "The Attic", it is revealed by one of Rossum's co-founders, Clyde Randolph, that the mainframe is Rossum's global communication and information processing system. The Corporation uses the mainframe in the place of conventional computer matrices.
The mainframe is unique in that it exists purely in the minds of those unfortunate enough to be sent to the Attic. Every Attic in every Dollhouse is linked via its inhabitants' neural connections. Each brain is kept fueled with adrenaline in order to ensure peak efficiency. In this way, the Rossum Corporation is able to both dispose with its enemies and transform them into localised CPUs.
As Rossum has assumedly dispensed completely with conventional computer systems, the mainframe holds every piece of information in Rossum's possession, and hence is the key to Rossum's destruction. Rossum maintains a localised processing and upkeep facility for the mainframe at their headquarters in Tucson, Arizona.
In "The Hollow Men," this facility was destroyed by Echo. Boyd Langton, having been wiped, was armed with explosives and told to detonate them. It is unknown what impact the destruction of the mainframe's hub had on the Attic or on Rossum's goals for global domination. As the future depicted in "Epitaph One" is set to occur, we can assume that Echo's attempt at destroying the mainframe, and hence the Rossum Corporation, was partially unsuccessful.