These connectors are all analog ways to get video in or out of your computer in a relatively low resolution. With the exception of HD-out, all of these connectors pass standard definition tv signals. All of these connectors (with the exception of composite) also look very similar to one another, so this is a section you'll definitely want to pay close attention to.
Composite output is an analog signal that outputs a standard definition TV resolution. It is compatible with all but the oldest TVs and is the lowest common denominator when it comes to television video inputs.
Composite connectors are getting pretty old, and I would highly recommend getting an S-Video or HD output if you are at all concerned with getting good picture quality on a TV screen. Most newer cards have switched over to at least S-Video (if not HD).
Plain S-Video is still very common on all but the latest video cards and is compatible with all but the cheapest televisions. S-Video outputs the same resolution as composite, but the signal is broken up somewhat differently, and with high quality cables you can see a noticeable difference in picture clarity
If you have a standard definition television, S-Video is a very good option. However component/high-definition connections are preferable if your TV supports them.
S-Video + Composite + HD
Several video cards come with a connector that looks very similar to S-Video, but has 3 extra pins in the center. This connector is a little confusing, because it can be two different connections entirely depending on what your video card manufacturer opted to do with it. On many cards it is an S-Video connection and a composite connection put together, while on other cards it is an S-Video connection and a component/high-definition output.
The S-Video/Composite version typically includes a small S-Video to Composite converter which will give you a standard composite output. To get S-Video, just plug in a regular s-video cable as usual, the extra 3 pins aren't used. This may require changing a setting or two in your video cards control panel.
For the HD/Component output version, the card will include a small breakout box to support the different connections.
VIVO (Video-In Video-Out)
VIVO connectors are very similar to s-video connectors as well. Not only do they have the extra 3 pins in the middle, but another 2 pins at the bottom for a total of 9 pins.
VIVO connections require some kind of breakout box or dongle to turn this single connector into multiple inputs and outputs. The available connections vary widely from manufacturer to manufacturer. Common configurations include a combination of S-Video and Composite input and output, or S-Video/composite input with HD output.
Some configurations also include the ability to use a regular s-video connector coming straight out of the VIVO port. Have a look around in the control panel for whatever video card you have (ATi CCC, or nVidia control panel) and see what kind of tv-out options it gives you.