First seen in the "The Corenthal Connection (Jessie's back)" the Black Box Device does not have a clear purpose or provenance yet. It superficially resembles a Geiger Counter. It has a window on the front with an idicator arrow, and a battery pack on the back, with a few wires sticking out from the black electrical tape wrapped around it. It originally had fuses on the top but Jeff states that he removed those.
Besides the video where it was found it has appeared in One step forward, Two steps back and May & June.
The Corenthal/Adam Roberts Notes appear to refer to a device being created to detect the presence of Sigma Radiation and an unusual collection of carbon, as found in the black cloth (which was also mentioned in the Corenthal Letters dated 7-22-1995 and 7-23-1995) that is described in those notes. It is speculated that this is that device, and that it is designed to detect Slenderman. The device could be potentially used as a warning mechanism or perhaps a locator for offensive purposes.
In the video Sigma, the device is speculated by Evan (following information given to him by Jeff) to measure sigma radiation, emitted from Slenderman. He tests two blood samples, one from him, one from Alex. According to the device sigma radiation is present in Evan's blood, but not Alex's. On August 11th Jeff tweeted that he, Evan, Vin, and Steph had all tested positive for Sigma radiation, according to this device.