The device center is a coordinate in 3d-space inside the eye tracker. From application development perspective, it’s best just to assume its a point on the eye tracker close to the center (in practice). Since it can very between tracker families, don’t see this as a well defined point. Internally we use it in our device calibration applications.
On Tobii Pro site, there is a very good overview of the different coordinate systems, please have a look there http://developer.tobiipro.com/commonconcepts/coordinatesystems.html
You see the track-box frustrum there, and now it narrows down towards the eye tracker device.