No offense BonyTony, but your link is to a SCART connector which is only used in Europe. In North America televisions do not use SCART connections.
In North America there are many ways to connect your Gamecube to your television:
1) RF connector (looks like regular cable connector)
2) RCA connectors (the standard 3 wires that come with the GameCube)
3) S-Video (requires separate RCA audio cable)
4) Component video (requires separate RCA audio cable)
All TV's that I know of have at least an RF connector input, most newer TV's have RCA input connectors, and higher end new TV's have S-Video or Component Video connectors.
Component video is the best quality-wise as it converts the video signal into three separate color signals (3 video wires), S-Video is the second best as it separates the black, white and color signals (1 video wire), RCA is third best as it puts the entire video signal through one wire and an RF connection is the worst because it puts all the video signals and the audio signal on the same wire.
As for improving quality if you currently use an RF connector any of these options will help you, if you are using RCA connectors S-Video or Component video (if your TV supports it) would greatly help.
As for improving resolutions, some games are programmed to work in 480i mode which is only availabe for Component and S-Video (I'm prety sure S-Video supports 480i). This 480i mode must be supported by your television for it to work, and the Gamecube must be "started up" in 480i mode (by holding the B button while it starts up).
Hope this helps,
Graham