The DVI cabling standard basically has the ability to transfer either a digital or an analog signal. That's why you can still get a projector with only analog ins to talk to a DVI port. The Mac Mini can output both the analog and the digital signal.
The DVI cables, however, come in varieties that only carry the analog signal (DVI-A) or only carry the digital signal (DVI-D), as well as a variety that carries both (DVI-I). Basically, it has the same connector on both ends, but some of the wires are missing on the inside in some versions. There's really no reason you should ever buy anything except a DVI-I cable, because that one always works, and doesn't really have any disadvantages, AFAIK. That is, you should never need a DVI-D cable.
Basically, you just need that both sides can talk on the same signal -- since the Mac Mini can do both analog and digital (that is, it transmits both signals out of the DVI port), this will always be true for any monitor. If the monitor can handle the digital signal (e.g. most newer LCDs), then it will use the digital signal, and if not, it will use the analog signal. The only time you would get in trouble is if, for some reason, you had a computer that only output an analog signal, and a monitor that only accepted a digital signal, for instance. Then, even with a DVI-I cable, the computer would be sending one signal and the monitor would be waiting for the other.
Does this help, or make any sense?
See
this too.
Oh, and in answer to your final question (1280x1024), yes, it does.
