Analogue delays use 'Bucket Brigade' chips to produce the delay. The method is not altogether dissimilar to the digital delay process in that incoming voltages are 'sampled', but, unlike digital, each sample is passed down the line to the next stage, much like a bucket of water being passed down a line of people trying to put out a fire at the other end. With multiple stages within one chip there will, therefore, be a delay before the signal reaches the output. Delay time can be changed by varying the oscillator that controls 'clock' time - this bit of circuitry determines how regularly a sample is taken.
The output signal will look slightly angular, which gets worse the longer the dalay is, so filtering is required at the end of the delay chain to smooth out the signal. The filter's cut-off frequency is usually fairly low to filter out 'clock noise' so the delayed signal sounds a bit dull and lifeless, but that's no big deal as it provides contrast to the original signal it gets mixed with (for dull, substitute 'warmth' in guitar mojo circles).
Digital delay uses an analogue to digital converter to convert voltages at its input into binary code. These numbers are stored, a microprocessor determines how long for, then they're sent along to the output where a digital-to-analogue converter 'reforms' the signal. Again, the signal needs filtering to smooth out the rough edges, though with high bit resolution less filtering is required (brighter sounding delayed signal, which some people find 'cold'). Also, delay times can be a lot longer than is possible with analogue delays (not unless you're content to put up with a really noisy, lo-fi delayed signal). Digital delays usually consume higher current (not practical to run them off a battery).
Tape delay is usually a loop of tape between two tape heads. One records while the other plays. Because it takes time for the tape to reach the playback heads there's a delay.
All the above is an oversimplification, but hopefully you'll get the idea.
