Connections that can be made with adapters are signals that tend to be basically the same but just use different types of connectors. For instance, HDMI and DVI share the same type of digital video signal so one can be adapted to the other easily. However, HDMI and VGA use completely different formats. One is a digital binary signal whereas the other is an analog sin wave signal. A device that can real one signal and actively convert it into the other format is needed in this situation.
There are several variations of DVI as well as derivative connectors based on DVI that have been used by different manufactures. However, the most commonly found versions found today are as follows. DVI-I carries both analog and digital signals. DVI-I can be single link or dual link. DVI-D carries digital signals only. DVI-D can also be single link or dual link. DVI-A that carries only analog signals. It does not have a dual link option. M1 P&D that carries analog and digital signals as well as USB serial information.
While both DVI and HDMI are each able to handle extended bandwidths, they do it in different ways. DVI uses a dual pipe connection where as HDMI simply increases bandwidth on the existing lines. So while they are compatible with each other in single link, HDMI does not connect with DVI on the dual link pins. You can still connect a dual link DVI device to HDMI. The devices will configure themselves appropriately, but be limited to single link speeds which still support 1080p.
DisplayPort was primarily designed as a replacement for VGA, DVI and LVDS which are common interfaces used in the computer/IT industries, while HDMI was developed by the Consumer Electronics Industry specifically for Home Theater applications. Toward this end, DisplayPort has certain enhanced features such as direct drive monitor design and single cable multi-function monitor connectivity. Some advantages of HDMI are the ability to support xvYCC color, Dolby TrueHD and CEC (Consumer Electronics Control) signals. Though both share powerful A/V features, DisplayPort was designed to complement HDMI, not to replace it.