HDMI is the most common Audio/Video connection used today. But with so many other Audio/Video connection standards, both young and old, there is a great need for adapters and converters.
So what converter do I need?
HDMI <-> DVI
HDMI and DVI only require a simple adapter to connect their digital signals! No need for expensive conversion boxes. You can also use a simple HDMI to DVI cable. note: DVI has no audio
HDMI <-> Displayport/mini Displayport
Displayport is digital but different, so you need a converter. The converters can be very small and even contained within a small adapter or cable and they can run off of the power provided by your HDMI or Displayport connection. Some require greater power and also include USB connections for more power. Some use an AC power adapter.
HDMI <-> VGA
HDMI is digital and VGA is analog and so you will need a converter box for this. Luckily these converters have been coming down in price and are becomming more "plug and play" than they used to be. The newer ones also support more VGA resolutions resulting in far better conversion quality.