A guitarist speaking of an analog delay mostly refers to a Bucket-Brigade Device (BBD) -based delay pedal.
Seen for a technical point of view, tape delays and oil can delays are also analog delay pedal.
Actually, the BBDs are some kind of analog/digital hybrids, because the signal is sampled discrete time based by the chip.
The difference between discrete time based sampling and digital sampling is, that the discrete time is only quantized temporarily, while the digital sampling quantizes temporarily as well as in the amplitude.
To keep this technical aspect as simple as possible, I would like to explain the way how an analog Bucket-Brigade Device delay works briefly: the so-called Bucket-Brigade device leads the voltage from one transistor/capacitor cell to the next. This works just like a Bucket-Brigade. According to a Bucket-Brigade, there is always some loss when the signal is transmitted again. Thereby, the signal alters depending on the transmission distance / the number of repeats.
Initially, the sound of an analog delay has been quite dull and somehow blurred. Because of the circuit, the maximum delay time was about 300 to 400 ms.
The headroom of many analog delays was limited what led to the fact, that the pedals their selves caused a slightly distortion.
But especially this provides the attraction of an analog delay because the signal fits perfect into the mix and melts with the sound of the guitar.
Today, there are numerous analog delay pedals with bright and percussive repeats which offer a maximum delay time of 1.2 seconds. If you are able to accept some hiss and a lo-fi sound, you can even get up t 4 seconds.
The short setting of the Moog Moogerfooger- 104M SD provides a sound that is as clear as the sound of a good digital delay pedal.
Because of the minimal variability of the delay time, many analog delays are able to provide modulation. Operating like a chorus pedal. A comparison that definitively makes sense, regarding the fact, that a chorus pedal actually is a delay too – only with a very short delay time and modulation.
By setting the modulation more intensive, you will get pitch effects because of the larger time shifting. If an analog delay pedal does not provide modulation, you can also create this kind of effect by turning the time pot back and forth slightly while you are playing your guitar. The modulation function of the pedal does in fact not work in a different way.
Another special titbit of analog delay pedals is oscillation. By turning the repeats knob into full clockwise direction, the signal will be built up until it overdrives. This sound can be used as a soundscape or as a crazy sonic element.
THE analog delay sound is not existing because the technology is not as limited as it has been initially. Nevertheless, the un-perfect, dirty and dark sound of an old analog delay still is one of the most famous ones.