The main reason for their popularity was because of apparent "intelligence", reflected in their ability to develop language skills.
Furbys can communicate with one another via an infrared port located between their eyes. Furbys start out speaking entirely Furbish, a language with short words, simple syllables, and various other sounds, but are programmed to speak less and less Furbish and more and more English as they "grow".
There was a common misconception that they repeated words that were said around them. This belief most likely stemmed from the fact that it is possible to have the Furby say certain pre-programmed words or phrases more often by petting it whenever it said these words. As a result of this myth, several intelligence agencies banned them from their offices.
A simple electric motor and a system of cams and gears close the Furby's eyes and mouth, raise its ears, and lift it off the ground in a faux display of mobility.
The originals are still popular with many hackers as they can be dissected and made to do interesting things. In particular, their advanced audio capabilities and various sensory interfaces make them popular with the circuit bending community.