The simple, unhelpful, answer is DVI-A is an analogue connector, DVI-D is digital.
There are also a few other variants.

• DVI-I which is the superset of both DVI-D and DVI-A
• M1-DA, which is DVI-I with USB too
• DVI-DL, which is DVI-D with the extra pins for a second data link (the DL means dual link), this provides support for very high resolution displays like the Apple Cinema 30" screen.

The signal carried by the DVI-A pins is interchangeable with VGA, a cable with both VGA and DVI-A plus will allow one to simply be adapted to the other.

The signal carried by the DVI-D pins is equivalent to the video part of the HDMI signal.

As I found out to my cost when I bought my first DVI lead, the plugs are physically different between the different types, DVI-A and DVI-D have two pins above and two pins below the wide pin, so a DVI-A plug won't physically fit in a DVI-D socket and vice-versa.

The various connector configurations are illustrated on this diagram.