HDMI cables are the latest technology in audio-visual connectors. With HDMI, one single cable delivers both video and audio from one component to another. HDMI enables the transmission of an uncompressed, full-bandwith digital signal from a high-definition source (like a Cable Box or Blu-Ray player) to an HD receiver (like a 1080p TV, or home theatre receiver).
An HDMI cable replaces either a set of component cables (5 plugs at each end---Red, blue, and green for video and red and white for audio), or a set of composite cables (3 plugs at each end---yellow for video with red and white for audio).
Either a compsite connection, component connection, or HDMI connection can be used to transmit and audio/visual signal from a device to a TV (depending on, of course, which connections your device and TV share). In terms of signal quality, the composite (3 plug) has the lowest quality and is unable to transmit a high definition signal. The component connection (5 plugs) is the minimum you need to transmit an HD signal, but its output is not as good as an HDMI cable. The HDMI transmits the highest quality signal of all, and is quickly becoming the standard of the industry.
One final note, a DVI connection is very similar to an HDMI connection, although the DVI connection is only for video, not sound. If you use a DVI connection, it will transmit a high-quality video signal, but you will still have to use a red/white cable set for the audio. Only HDMI does both sound and video in one cable.