There are three (and a half) typea of analog reverbs, and two types of analog delays.
Delays first:
The old type is called tape delays. The use tape in a loop. Roland space echo is the most famous type there.
Second type is known as analog delays. The basically use a chain of condensers that delay the signal. The delay time depends om the amount of condensers and the samplerate. Yup, samplerate. They sample. So they are kinda half-digital.
Reverbs:
First there was echo-chambers. Big rooms built to have a long and nice echo. You put a speaker in one and and mics at the other. Obviously hugely expensive.
Then came the spring reverb. Uses springs. Sounds very special, metallic and eh, springy. Sounds great on guitar but not much else.
Then came the plate reverb. Sounds great, smooth, fantastic. About as big as a bed frame, and needs to be kept where nobody risks bumping into it while recording.
The last gasp was the gold foil reverb. Basically a miniturized plate reverb using gold foil instead of a steel plate. Wasn't very popular.
So, unless you have loads of dough and space, may I suggest you drop the "digital sounds like shit" attitude and buy a digital reverb? Trust me, it's gonna sound much better than a spring reverb.