Simple echo repeats aren't the only thing current delays can do because over the years, a lot of creative work has gone into their development to make the delay sound more diverse and create entirely new sonic landscapes.
You can set the delay time of the echo to predetermined note values and integrate the repeats almost like a second instrument into the mix. The Edge from U2 perfected this way of playing with delay. In this style, the echo signal matches the original signal in volume. For using tempo delay sounds, one thing is quite handy: a tap function, where you can tap the tempo with a footswitch. Most delay effects available today have such a tap button.
You can only create this variant with a stereo delay effect. In this style, the echo repeats move from left to right: the first repeat on the left, the second on the right, the third back to the left, and so on.
Multiple delays with different delay times are combined here, resulting in an intricate delay sound with a special rhythmic structure. Guitarists like David Gilmour or The Edge used to achieve this by chaining multiple delay effects in a row. Today, many digital delays have a specific multitap algorithm for this sound.
As the name suggests, the echo effect is played backward. What used to be a complex action with tape delays is now a standard function in many digital devices.
In this delay style, a pitch-shift effect, i.e., a change in pitch, is applied to the echo repeats. The result is a very distinctive sound that can even create small melodies depending on the settings. It's also possible to assign the additional tones of the pitch shifter to a key (harmonizer effect).
With this function, the echo is essentially "frozen": you play with delay and press the hold button, which keeps the current sound unchanged. This can then serve as a basis for your own playing, making it ideal for ambient sounds and song intros.
Granular synthesis involves capturing and playing back short fragments (under 50 ms) in a distorted manner. This type of sound generation is heavily used in synthesizers, but there are also delay effects with this very distinctive sound. The echo repeats are not really controllable in this case and do not produce a rhythmically usable delay sound. Instead, they create more like sound clouds that can be used especially for atmospheric sounds.