Atrac is unique from many other forms of audio compression in that it's not a pure-maths based technology like Mpeg or PCM. It's mostly maths, but the designers also used a multitude of human subjects to help them 'prune' the waveform, removing frequency bands that humans can't hear anyway.
Although this damaged its quality, it increased the compression ratio dramatically, one engineer is quoted as saying 200%, up to a 5-to-1 ratio. It is also therefor not as uniformly lossy as others, though still a lossy format. The difference could be explained that Mpeg losses bits from all over the waveform, whereas ATRAC primarily loses bits from specific wavelengths.
"Why didn't they just use MP3?" is what a lot of people ask. Well, there are a lot of reasons. Although Mpeg wasn't a ratified standard in those days, it still existed in the unofficial format (which is nearly identical to how it exists today), so they could have used it. The problem is that MP3 requires a lot of horsepower to decode -- at least a 486/133. And in the early '90s that kind of power was both expensive and power-hungry. By comparison, an Atrac encoded stream can be decoded on machines with scarcely more power that a midrange 386.
The ATRAC standard also has the design benefit of being forwards compatible; an MD recorded with Atrac 5 can be read just fine on a deck that was only deisgned with Atrac 1. That's because the advancements with successive generations affect only the encoder technology, not the decoder; the decoder just uses set algorithms.