An Analog Shift Register (ASR) is a cross between a Sample & Hold module and a Bucket Brigade Delay (assuming you already know how those work). When initially triggered, it samples the incoming voltage, and presents that at its first output. On the second trigger, the incoming voltage is sampled again with this new voltage presented at the first output, while the original voltage is now moved to a second output. This game of “telephone” is passed along for as many stages as the ASR has – traditionally three or four.
These voltage typically are sent to different oscillators, which will then play intervals and chords based on what notes you feed in one at a time. The output is akin to an echo with two or three repeats, but rather than being spaced at equal intervals of time, the “echoes” are triggered at the timing of your choosing, creating what has been called an arabesque pattern.
Although the history is a little hazy, it appears the first ASR was created by Fukushi Kawakami for composer Barry Schrader in the early 70s. It was then made available as a Serge module in the mid-70s. A few companies and DIY designers make ASRs available as modules today, including Verbos in their Random Sampling module and Elby Designs, using Ken Stone’s design.