There are basically two kinds of guitar delay pedals available: analog delay pedals and digital delay pedals. Both have their strengths, and can be at home on a guitarist’s pedalboard.
Analog delay pedals usually rely on a bucket-brigade device (BBD) chip that sends the analog signal through a series of capacitors, one step per clock cycle. The repeats of analog delays tend to get a bit warmer and darker and a bit more diffuse (broken up) with each step, which imparts a very particular sonic signature. Due to the limitation of BBD chips, analog delay pedals tend to offer shorter maximum delay times than digital delays.
Digital delay pedals use digital signal processing (DSP) chips to create their echo effects. This means that they can be as colored or transparent, or long or short, as the designers choose to make them. They offer more flexibility than analog delay pedals, some even offering MIDI control. However, your guitar signal must be converted to digital at the front end of the pedal, and then back to analog at the output, and not all digital delay pedals have had the best A/D and D/A converters. These days, however, modern pedals offer 24-bit resolution and quality conversion.
Which type of delay pedal you prefer is a matter of taste. Analog delays can sound extremely musical and pleasing, and some guitarists don’t like their signal to be converted to digital if they can help it. But digital delays offer immense amounts of flexibility — including analog delay emulations, precise delay time control, preset memories, and other features.
But why choose? Many guitarists place both their favorite analog and digital delays on the same pedalboard!