The main point of a HDMI cable is to run high def video and audio on one cable. But you can also use it to run just video(or audio if you wanted, I guess). Example, my surround sound receiver is older and doesn't have any HDMI ports. So I use a HDMI cable to run the video from my PS3 to my HDTV and then I use a fiber optic cable from the PS3 to the receiver for audio. I could use the cable for both video and audio, but it would only go to the TV and I would not be using my surround sound.
I've never run my computer to my TV using an HDMI cable, but to meet the HDMI spec requirements I would assume the video card would output both audio and video signals.