A digital synthesizer is basically a synthesizer that uses digital methods to generate sounds.

"Digital synthesizers" have been around since the late 1950s, when RCA introduced the Mark I synthesizer to the world. The "synthesizer" consisted of a variety of analog components (oscillators, etc.) - the catch is that the components were digitally controlled. In order to use this program, you had to tediously program in a piece on punchcards. This severely restricted its use.

Some commercial digital synthesizers came out in the late 70s for musicians with deep pockets (notably the New England Digital Synclavier workstation and the Fairlight sampler), but it wasn't until the Yamaha DX7 came out that people took notice. The Yamaha DX7 packed an amazing amount of synthesizer power (realistic instruments, nice sounds) for a $2000 price tag. The result was a huge success that redefinied the shape of the synthesizer market.

Digital synthesizers are very similar, technique wise, to analog ones. Usually, they start out with a sound source (anything from a sample to a computer-generated wave) and process them through various digital algorithms. The result is often more realistic (due to the samples), but many complain digital synthesizers sound "colder" than analogs.

Some synthesizers have been made using digital synthesis methods that do not rely on samples. The Yamaha DX7 is a good example of this. It used digital methods to do a type of synthesis called frequency modulation (FM) in much greater quantities and with much greater precision than was possible with an analog synthesizer of the equivilent size. The Roland D-50 is another example. It allowed you to select either short "PCM" samples or digitally-generated waveforms and process them through digital filters and digital VCAs. However, today's digital synthesizers primarily use samples to generate their sound.