A delay effect takes a signal (the "input signal") and produces a repetition of the signal (the "output signal") with a delay in time (the "delay"). Practical delays will also change – usually reduce – the amplitude of the output signal from the amplitude of the input signal (the "decay").
A delay effect is an effect the output of which is the input signal repeated at least once with some delay in time and usually with some decay in amplitude.
The input signal, also called the "dry signal", may or may not be added to the output signal. The output signal (usually just the repeated, delayed, decayed part) is also called the "wet signal".
Examples of delays and sound samples
Some delay effects allow the wet signal to be fed back into the output. Each time the delayed output signal is fed back into the input of the delay effect, another signal repetition is created. Such "feedback" can theoretically continue indefinitely and in effect creates an "echo", where the signal is repeated many times with ever decreasing amplitude.
Delay and decay sweeps
Some delay effects allow for a gradual change in the amount of time delay ("delay sweep") or of the decay ("decay/amplitude sweep"). Varying the amount of delay in a simple delay with a short delay and no decay creates a flanger.
Some delay effects may contain more than one simple delay unit and thus the output signal may be the result of several signal repetitions with delay and decay amounts that are independent with each other. Such effects may contain delay units that work in parallel or in a chain, may allow feedback or sweeps at different places of the delay chain. Some delay chains may allow the output of each delay unit to be "tapped" and taken to the output before it is fed into the next delay unit in the chain, thus resulting in a "multi-tap delay effect".
A "slapback delay" is simply a delay effect with a very short delay time. A slapback delay has a delay time of, say, 50 ms to 250 ms (different sources point to different times). A slapback delay will usually have a single repetition (no feedback) and non-varying delay and decay amounts (no sweeps).
Short delays with large decays are very common in music production. When the delayed repetition is quick and quiet, it is not heard as a separate repetition, but rather changes how the original signal sounds. See, for example, this post on Fuller guitar solo with a delay bounce. The post uses a delay that is longer than the ones described in the previous paragraph – 300 ms or 450 ms – but these repetitions, if audible, hide behind the drums. The decay is large. Only 20-30 percent of the original signal amplitude is preserved in the repetition. The repetition itself is not heard as a separate repetition. It creates a fuller guitar solo with a reverb feeling (the final mix of this song does not use reverb on the guitar).