Here it is, the daddy of Delay pedals: The Electro-Harmonix Deluxe Memory Man.
I've been trying to get my grubby little paws on one of these for some time now and now that I have I must say I am not one bit disappointed, in fact I'm over the moon with it.
This pedal is an "analogue delay". The technicalities of analogue vs. digital are; that instead of a digital memory buffer to store audio and release it after a set period of time (digital) the "vintage" method employs dedicated integrated circuits called BBD chips (MN3005 and the rest of the family). Why are these better sounding (to me at least) than digital delays?
The repeats are not perfect, spectrally speaking: each time the output is fed back into the input (a process known as Feedback) it loses some high frequency detail. This process results in more "natural sounding delays" that are technically just lacking in high frequency detail. I believe this plays a major role in the reasoning behind BBD based delays being favoured the world over. That and the phase deviations above 10kHz.
If you're interested in the specifics of the analysis in relation to frequency response, there's a rather good paper from the fine people of Stanford here.
Otherwise, here's a quick sample: