Can't take the full credit for this one, info comes from a magazine (OXM) that makes the fanboys in the graphics forum look tame. However, I've added my own comments as well 
VGA - PC Monitor, LCD, Projector, Plasma, CRT - Best Option
VGA breaks the signal into 3 channels, Red, Green & Blue. It also has TWO synchronisation channels as well. Use VGA for anything that accepts it in most cases. Of course in the case of LCDs and Plasmas, the signal has to be converted to digital. The quality of the converter plays a big part, and in some cases Component will look better (although sometimes that can be due to a different colour temperature setting on the TV). VGA is progressive and supports resolutions higher than the 360 is officially capable off.
Component - LCD, Projector, Plasma, CRT - Second Best Option
Basically works the same as RGB SCART in principle, however it has a major advantage - high resolution support. Component supports interlaced, as well as progressive - it also supports higher resolutions (such as 1080i).
On a lot of TVs this can offer better image quality than VGA, either due to a better converter, or different default Colour/Brightness/Contrast settings.
RGB SCART - CRT - Third Best Option
The signal is broken up into three channels - Red, Green & Blue. A composite feed is also used to synchronise the three channels.
The signal is interlaced, this means it is NO GOOD FOR PROGRESSIVE SCREENS. It'll work, but image quality won't be as good as it should be. Progressive screens include 100Hz CRTs, LCDs, Plasmas, PC Monitors. Resolution appears to be fixed at 720x288 (720x576 effective over 2 frames). No HD gaming to be found here.
S-Video - Don't use
Chances are you won't use this unless you're American, or restricted to it as the best connection you can have. S-Video is like composite, except it splits the signal into two channels: Chrominance and Luminance. Its better than composite, but no match for RGB SCART, Component or VGA.
Composite - Don't use - Last Resort
Rubbish, keep away from this. Basically, everything goes down a single cable to the TV. Your stuck at the standard PAL or NTSC res (depending on region), its interlaced, and generally rubbish. Last resort only
Composite is also used by default with non-RGB SCART sockets, and when using one of those 3 phono -> SCART adapters. Watch out for this.
DVI / HDMI - Not currently used by any console (just relays a VGA signal)
In terms of all current consoles, using DVI or HDMI is no different than component. The SAME signal is sent through the port as with component. Any difference in image quality is due to the TV set (for example, 1:1 pixel mapping on some LCD TVs), or the placebo effect.
Ultimately if any console outputs a genuine digital signal as supported by DVI & HDMI, then the image quality would be superior to all other connection types. For one you would be skipping two conversion steps (Digital > Analogue > Digital), but you would also avoid all the interference inheriant with analogue signals.

VGA - PC Monitor, LCD, Projector, Plasma, CRT - Best Option
VGA breaks the signal into 3 channels, Red, Green & Blue. It also has TWO synchronisation channels as well. Use VGA for anything that accepts it in most cases. Of course in the case of LCDs and Plasmas, the signal has to be converted to digital. The quality of the converter plays a big part, and in some cases Component will look better (although sometimes that can be due to a different colour temperature setting on the TV). VGA is progressive and supports resolutions higher than the 360 is officially capable off.
Component - LCD, Projector, Plasma, CRT - Second Best Option
Basically works the same as RGB SCART in principle, however it has a major advantage - high resolution support. Component supports interlaced, as well as progressive - it also supports higher resolutions (such as 1080i).
On a lot of TVs this can offer better image quality than VGA, either due to a better converter, or different default Colour/Brightness/Contrast settings.
RGB SCART - CRT - Third Best Option
The signal is broken up into three channels - Red, Green & Blue. A composite feed is also used to synchronise the three channels.
The signal is interlaced, this means it is NO GOOD FOR PROGRESSIVE SCREENS. It'll work, but image quality won't be as good as it should be. Progressive screens include 100Hz CRTs, LCDs, Plasmas, PC Monitors. Resolution appears to be fixed at 720x288 (720x576 effective over 2 frames). No HD gaming to be found here.
S-Video - Don't use
Chances are you won't use this unless you're American, or restricted to it as the best connection you can have. S-Video is like composite, except it splits the signal into two channels: Chrominance and Luminance. Its better than composite, but no match for RGB SCART, Component or VGA.
Composite - Don't use - Last Resort
Rubbish, keep away from this. Basically, everything goes down a single cable to the TV. Your stuck at the standard PAL or NTSC res (depending on region), its interlaced, and generally rubbish. Last resort only

Composite is also used by default with non-RGB SCART sockets, and when using one of those 3 phono -> SCART adapters. Watch out for this.
DVI / HDMI - Not currently used by any console (just relays a VGA signal)
In terms of all current consoles, using DVI or HDMI is no different than component. The SAME signal is sent through the port as with component. Any difference in image quality is due to the TV set (for example, 1:1 pixel mapping on some LCD TVs), or the placebo effect.
Ultimately if any console outputs a genuine digital signal as supported by DVI & HDMI, then the image quality would be superior to all other connection types. For one you would be skipping two conversion steps (Digital > Analogue > Digital), but you would also avoid all the interference inheriant with analogue signals.
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