ADAT is an eight-track digital tape recorder system. Or to be more correct it used to be tape. Now the ADAT standard is being applied to harddisk technology and the eight track limit is overcome. Modern ADAT systems can record 90 minutes of 24 48 kHz tracks on 20 GB IDE harddisks (The Alesis HD24). The system is designed for small studios and high-end home studios. With this in mind, the units are compact and easy to operate. The more modern ADATs have more choices and are a bit more complicated to operate. For less demanding tasks, Alesis still sells the tape ADATs.
Fun fact: The tape is the same as Super VHS tapes.

Recording eight different tracks digitally was a very new feature for home studios when ADAT arrived in the beginning of the 1990's. ADAT also created its own standard for transporting sound via optical cables, sporting its own data protocol. This protocol is in use today and is called the ADAT Optical Transfer Protocol. One optical cable can transfer eight 20-bit, 48 kHz digital soundtracks. Today, you can say that a system with that protocol is an ADAT system.

The ADAT system also lets you chain multiple ADAT recorders together. The system uses ethernet (!) or a serial interface for letting the recorders talk to each other. In total, 16 ADAT units can act together in this way forming a substancial recording base. This was the beginning of the Modular Digital Multitrack technology. The digital recorders before this was usually 1" 32-track Denon reel tape technology.

For a picture of the insanely cool Alesis HD24: http://www.alesis.com/products/hd24/HD24.jpg