The DVI cable is for Digital Video Input - a better way of connecting a TFT monitor. But you need a DVI input on the monitor, and a DVI output on the graphics card.
If you don't have a DVI output on the graphics card then there is no way to get any of the benefits (converters will not suffice) - unless you buy a new card.
The DVI connector is slightly bigger than the normal VGA connector, and nothing like any other connector on a graphics card, so it's unmistakeable.
The reason it's potentially better is that it transfers digital signals directly to the monitor, without converting then to analogue VGA signals, for the monitor to convert back again to digital. This onlt applies to TFT monitors.