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The good news is that the transition to digital is easier to make because different DVI connectors are backwards compatible with analog signals and cheap adaptors mean compatibility with VGA ports. The bad news is you need to know which DVD cables do what.
There are 2 factors that differentiate DVI cables: the types of signal carried, and the presence or absence of Dual-Link capability. Cables can be DVI-A, DVI-D, or DVI-I, meaning they carry signals which are, respectively: Analog only; Digital only; or, in the case of Integrated, both. DVI-D and DVI-I cables can also be Dual-Link. This allows them to carry more data, used by screens with performance higher than a 1920 x 1080 resolution at 60Hz refresh rate and 24 bit colour depth.

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The picture at right shows a DVI-I, Dual Link, female connector. It has holes instead of pins; the pins are on the male connector. The monitor and graphics cards have female ports, and cables have male plugs at both ends. The red squares show the holes for the analog signal, which is how you can tell a DVI-I from a DVI-D. The green square shows the extra 6 holes for the Dual Link signal, which is how you tell a Dual Link from a Single Link DVI connector. The picture on the left shows which holes and pins would be absent from a DVI-A, which are always single link. Note that while the female connectors have a cross shaped hole; the male ones usually have just a horizontal tab.
Some argue that the –D and –A suffix cables do their specific jobs better, and this is probably true at some level. Particularly if you have the kind of equipment that makes the cable costs negligible. But, for someone in the process of upgrading, DVI-I is your friend.