Analog Delay
Imagine a line of laid back, hippy monkeys, each with a guitar just like yours. They're sitting on little stools in a cunning arrangement such that each monkey can only hear the monkey sat before him. The first monkey hears what you played, turns to the next monkey and plays it back to him. He then turns, and plays to the next monkey. At each stage, little mistakes are introduced due to the hippy monkeys' inherent laid-back-ness. At the end, the last monkey plays, and his guitar is connected to the output.
The monkeys are in a big big circle, and the first monkey, as well as hearing what you're playing, can also hear the last monkey - but not very well. There's a baffle between them, which cuts down how much he can hear. The first monkey plays both what you're playing and what he can hear of what the last monkey is playing, and this is passed on as before.
Now imagine the monkeys are tiny, as before, with tiny guitars and tiny stools and a tiny baffle. They're in a box. A knob labelled "feedback" controls the baffle. Then, kapoosh! The monkeys disappear and are replaced by a Bucket Brigade chip, and off we go.
Digital delay
Exactly the same as analog delay, but there are many more monkeys, and they're all actuaries and very precise. Some people miss the laid-back-ness of the hippy monkeys, but many like knowing that little actuary monkeys are taking very good care of what they're playing. Some of the monkeys know special tricks, like making what they play come out of the output even though they're not the last monkey.