HDMI video cables carry audio and video signals in a single cable, which makes them much easier to connect.
Note that with HDMI,
HDCP applies. With HDCP, certain connections require support of this specific security protocol across all aspects of the connection. This typically is only a problem when you have something
in between your source (i.e., the Wii, a Blu-ray player, a DVR, a video tuner box, cable box, satellite box, etc.)
and your display (i.e., your television). This
device-in-the-middle is generally used to allow you to switch your display between
several HDMI sources. If you have one of these devices-in-the-middle, you typically know it, because generally they're only used by video-geeks. However, having said that, I know folks who have a setup like this, who very definitely are
not video-geeks. They are folks who purchased a full system (including a Blu-Ray player) from a high-end video store, including installation that included connection of their HD DVR from their cable company, so all the geeking was done for them. However, a software update from their cable company, a few months after they purchased the system, resulted in nothing but an error message on the television -- you guessed it -- all because of HDCP.
This is the only significant down-side of HDMI, as compared to component video (plus TOSLINK or SPDIF for audio). To good news is that if you're hit with this problem, you can just switch to component video (plus TOSLINK or SPDIF for audio) -- the difference in quality isn't that noticeable.
Now I'm confused... I just bought a 42 inch LG LCD TV and my son is getting Wii from Santa... so now I need to by the above mentioned Wii Component video cable to have a good picture?
No -- you need component video (or HDMI) to have a
great picture. You can get a
good picture with the standard cables.
I don't know much about Wii's, but component video cables are cheap. I've got a couple of dozen spares in my basement. (I'm not kidding. We cleaned out closets this weekend, and collected all cabling together. I could open my own Radio Shack, at this point.

) Is there something about the connections on the Wii that don't allow you to use standard component cables?