HDMI is today the most high-end kind of video connection available and you should use it if the parts you are connecting support this connection. You will find this connection on the latest DVD players, Blu-Ray players, HD-DVD players, HDTV sets and probably on the latest digital cable/satellite converters.
This connection works 100% in digital mode, is capable of transferring higher resolutions than DVI – including resolutions not commercially released yet –, features a copy-protection mechanism called HDCP (High Bandwidth Digital Copy Protection) and transfers digital audio signal on the same cable, eliminating the need of an extra cable for connecting audio – if your devices support this feature.
We have already written a tutorial explaining in-depth how HDMI works, so you may want to read it if you are interested on HDMI.
On the pictures below you can see some examples of HDMI use.

click to enlarge
Figure 32: HDMI output on a DVD player.

click to enlarge
Figure 33: HDMI inputs on a HDTV set.

click to enlarge
Figure 34: HDMI cable.
As we mentioned you can easily convert DVI-D and DVI-I to HDMI thru a cable or adapter. If you want to connect your PC to your HDTV set you will need to use this kind of cable or adapter, see Figures 35 and 36.

click to enlarge
Figure 35: DVI-to-HDMI cable.

click to enlarge
Figure 36: Two DVI-to-HDMI adapters.