I believe they perform the signal amplification in the digital domain, with the signal being broken down into 0's and 1's. They have a much higher sampling rate than redbook CDs and some offer analog and digital inputs. The power supplies appear more efficient than those found in analog designs. They are lighter, run cooler, and are generally less noisey than their analog counterparts.
I recently bought Panasonic's SA-XR45 digital, 6-channel receiver, which can be found for around $300 on the Internet. I must admit that overall I prefer it to some highly-rated tube and solid-state analog amps I've had in the same system with my Klipsch Chorus. I see a very strong case for digital amps in the future.
I still use tube-MOSFET hybrid monoblock amps in my main reference system, but in the future I may try digital amplification based on my positive experience with the Klipsch/Panasonic system. BTW, Newform Research recommends the Panasonic XR-45 for bi-amping their R645 speakers (used in conjunction with a Behringer digital crossover)...
So when Theo says, "THE PRICE!!!", I'm not sure which way he means it!!! Some digital amps (TACT, for example) are expensive, but the technology is rapidly trickling down...