Delay line is a device that repeats the value of its input as it was some short time ago.
The "pure" delay line should remember all history of its input changes and "replay" it with delay. "Real" delay lines are expensive to build and seldom used (they have been used as memory devices in the past). When this is actually required, delay line may be implemented by passing the signal through some volume (quartz, mercury), usually first converting electric signal into accustic signal (sound waves are much slower so longer delays are possible).
Usually in real world digital electronics delay line is approximated by some other circuit that under given conditions of operation has the similar behavior (frequently RC circuit is used). Such line delays front and tail of the signal but has no memory, so changes shorter than the delay time would not be "replayed" on the output. Real world logic gates also have some very short delay on they own. A delay line with memory can be built from the shift register.
Delay line is not strictly a logic gate but it is required to explain how most of the statefull (memory-capable) circuits work.
Combining delay line and NOT element, it is possible to build the simple generator. Delay line determines the frequency of the generated signal. Results of connecting input and output without delay line depends on how the NOT element is implemented. If the circuitry of the NOT element has a strong enough tendention to delay the signal on its own, the generator still works, producing high frequency signal. However it is also possible that due negative feedback the circuitry will stop working as a digital logic gate and will set voltage lying between determined 0 and 1 levels int the output. The closed ring, containing the of odd number of NOT elements, has stronger delay capabilities and are more likely to keep working as generator (the ring containing even number of NOT elements will enter one of the two possible stable states). Simple real world generators usually approximate delay line by RC chain and typically contain a ring of three NOT gates. To make a generator, the number of inverters in the ring must be odd. A ring will even number of inverters would permanently enter one of the two possible stable states.
Devices like triggers or shift registers require to form pulses matching front or tail of the relatively slow changing input signal. Such pulses can be formed using gate combination that produces 1 when values at its inputs are different. Then on input can be connected to the source of the signal directly, and another can be connected through the delay line. The circuit will generate the pulse lasting till the signal from the source passes the delay line.
The most obvious candidate for this task is a XOR gate that is defined as a comparing element. Device with the delay line and XOR gate produces pulses, matching both front and end of the input signal.
However circuit usually should only produce one pulse, either at time of the positive (0 to 1) or negative (1 to 0) change of the signal in its input. Such circuit can be built using delay line and inverter together with AND element. It is an important component of counting triggers.
Edge detectors are described in more detail at .