Will I really notice any difference if I run an S-Video cable from my DVD player to my TV?
Yes - especially if you have a flat rather than curved picture tube. The S-Video signal is noticeably clearer and more defined - I believe that the technical explanation has something to do with color separation or something.
You will not, however, notice a difference between S-Video and Component video unless you have an HDTV.
I assume by standard plugs you mean RCA connections. And yes, you would. But an even better solution (if your TV has it as an input) is to use component cables. It’s the set of three cables, a red, blue, and green one.
Maybe. The only answer this question without further information.
Basically your composite video signal (the “standard” video) will need to be decoded either by the DVD player or the TV set. It will need further decoding withing the TV as well before it is finally displayed. There’s a lot of hype concerning s-video and component video, but if you have an inexpensive DVD player and a really good TV, it’s possible that you’re better off running composite video signal and letting the TV decode it.
Now, if you have a great DVD player and a crappy TV, you’re better off buying a new TV. 
Not true at all. S-Video is still interlaced NTSC, whereas component is progressive-scan. S-Video is about 350 lines of resolution, as I recall, and component is 480 minimum.
If your TV has component inputs, and your output device has component outputs, you’re much better off using them than any other connection method. Even if your TV isn’t HDTV - which just means it’s not capable of resolving HD resolutions of 720 lines progressive or 1080 lines interlaced. Most TVs that have component inputs can show 480p.
And progressive scan is much better than interlaced. Even displaying a regular TV signal if your output is coming from a device that has a line doubler built in. Even non-HD digital cable with progressive outputs looks damned close to DVD quality on a good channel.
The difference between S-video and composite is that the chroma and luminance signals are on separate conductors in an S-video cable. Other than that, the signal is identical. Keeping the two signals separate means you don’t have to use a chroma separator in the receiver, which can degrade the image.
A bit of clarification.
I have a good TV, a Sony WEGA, I have a cheapy Toshiba DVD player. The DVD player came with the 3-wire RCA connectors that connect to the TV, they are red, white and yellow. The instructions for my TV say if I want to use the S-Video connection, it would replace the yellow connection.
I suppose I could spend a couple bucks for an S-Video cable and see for myself if there is any noticeable difference. The way it is set up now seems to be just fine though.
S-Video will definitely provide a better picture. The yellow composite cable you’re currently using combines all of the video info in one signal. S-Video sends 2 different signals, one for chroma or colour and the other for luminence or brightness.
I don’t buy it. I have a Sony WEGA 27" with component inputs, and I see zero difference between the S-Video and Component signals (i’ve even done A/B tests with them, hot-swapping the two cables to compare). I should pount out that my source (Sony PS2) is not a progressive device.