Single Link DVI-D compared to HDMI has equal quality, except HDMI does allow you to send sound to the monitor/TV. Both are digital.
Dual Link DVI-D allows for 120hz, which HDMI cannot do.
DVI-I allows for both DVI-D or DVI-A. So it can be used with a converter to support VGA connections as well, DVI-D cannot. Some if not most DVI-I slots on a video card support Dual Link, which allows for higher resolutions and refresh rates.