'Digital Gate-Array IC', whatever that means.
According to Kenko:
It means the converter’s unique circuitry maintains signal integrity between the camera body and lens.
From that one could surmise that Kenko DG or DGX products are made to allow communication between the camera and lens.
The DG series of products is designed to work better with digital DSLRs and the DX only lenses often used with them. The main difference would probably be the lack of an aperture control ring on DX lenses.
If you are using a lens that relies on an electrical signal from the camera to set the aperture value of the lens this would be a necessity to allow normal operation of the lens' aperture. It also means that without the electronic connection your camera will not electronically detect the lens. At least some of the pre-DG products apparently also had pass-through electrical connections.
If your camera is unable to shoot in Manual exposure mode without communication from a lens, you would not be able to use "dumb" extension tubes that do not allow your camera to communicate with your lens.
In the case of teleconverters, the 'X' also means the signal from the lens is translated to reflect the changes to focal length and aperture that occur when attached to the TC. For example, if I shoot at maximum focal length with maximum aperture using an EF 70-200mm f/2.8 lens and a Kenko Teleplus Pro 300 AF 2.0X DGX, it reports a focal length of 400mm and f/5.6 to the camera.